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All Textbook Solutions for Calculus Volume 3

[T] The fence vector F acting on a proton with all electric charge of 1.610-19C (in coulombs) moving in a magnetic field B where the velocity vector v is given by F=1.610-19(vB) (here, v is expressed in meter per second, B is in tesla [T] , and F is in newtons [N] ). Find the force that acts on a proton that moves in the xy -plane at velocity v=105i+105j (in meters per second) in a magnetic field given by B=0.3j ).[T] The fence vector F acting on a proton with an electric charge of 1.610-19C moving in a magnetic ?eld B where the velocity vector v is given by F=1.61019(vB) (here, v is expressed in meters per second, B in T, and F in N ). If the magnitude of force F acting on a proton is 5.91017N and the proton is moving at the speed of 300m/sec in magnetic ?eld B of magnitude 2.4T , find the angle between velocity vector v of the proton and magnetic ?eld B . Express the answer in degrees rounded to the nearest integer.[T] Consider r(t)=cost,sint,2t the position vector of a particle at time t[0,30], where the components of r are expressed in centimeters and time in seconds. Let OP be the position vector of the particle after 1sec . Determine unit vector B(t) (called the binormal unit vector) that has the direction of cross product vector v(t)a(t), where v(t) and a(t) are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after t seconds. Use a CAS to visualize vectors v(1),a(1), and B(1) as vectors starting at point P along with the path of the particle.A solar panel is mounted 011 the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) A(8,0,0),B(8,18,0),C(0,18,0), and D(0,0,8) (see the following figure). Find vector n=ABAD perpendicular to the surface of the solar panels. Express the answer using standard unit vectors. Assume unit vector s=13i+13j+13k points toward the Sun at a particular time of the day and the flow of solar energy is F=900s (in watts per square meter [W/m2] ). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors F and n(expressed in watts). Determine the angle of elevation of the Sun above the solar panel. Express the answer in degrees rounded to the nearest whole number. (Hint: The angle between vectors n and s and the angle of elevation are complementary.)Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 1. First, write down two vectors, v1 and v2, that lie along L1 and L2, respectively.Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 2. Find the cross product of these two vectors and call it N. This vector is perpendicular to v1 and v2, and hence is perpendicular to both lines.Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 3. From vector N, form a unit vector n in the same direction.Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 4. Use symmetric equations to find a convenient vector v12 that lies between any two points, one on each line. Again, this can be done directly from the symmetric equations.Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 5. The dot product of two vectors is the magnitude of the projection of one vector onto the other—that is, AB=ABcos, where is the angle between the vectors. Using the dot product, find the projection of vector v12 found in step 4 onto unit vector n found in step 3. This projection is perpendicular to both lines, and hence its length must be the perpendicular distance d between them. Note that the value of d may be negative, depending on your choice of vector v12 at the order of the cross product, so use absolute value signs around the numerator.Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 6. Check that your formula gives the correct distance of |25|/1981.78 between the following two lines: L1:x52=y34=z13L2:x63=y15=z7.Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 7. Is your general expression valid when the lines are parallel? If not, why not? (Hint: What do you know about the value of the cross product of two parallel vectors? Where would that result show up in your expression for d? )Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 8. Demonstrate that your expression for the distance is zero when the lines intersect. Recall that two lines intersect if they are not parallel and they are in the same plane. Hence, consider the direction of n and v12 . What is the result of their dot product?Figure 2.73 Industrial pipe installations often feature pipes running in different directions. How can we find the distance between two skew pipes? Finding the distance from a point to a line or from a line to a plane seems like a pretty abstract procedure. But, if the lines represent pipes in a chemical plant or tubes in an oil refinery or roads at an intersection of highways, confirming that the distance between them meets specifications can be both important and awkward to measure. One way is to model the two pipes as lines, using the techniques in this Chapter, and then calculate the distance between them. The calculation involves forming vectors along the directions of the lines and using both the cross product and the dot product. The symmetric forms of two lines, L1 and L2, are L1:xx1a1=yy1b1=zz1c1L2:xx2a2=yy2b2=zz2c2 You are to develop a formula for the distance d between these two lines, in terms of the values a1,b1,c1;b2,c2;x1,y1,z1; and x2,y2,z2. The distance between two lines is usually taken to mean the minimum distance, so this is the length of a line segment or the length of a vector that is perpendicular to both lines and intersects both lines. 9. Consider the following application. Engineers at a refinery have determined they need to install support struts between many of the gas pipes to reduce damaging vibrations. To minimize cost, they plan to install these struts at the closest points between adjacent skewed pipes. Because they have detailed schematics of the structure, they are able to determine the correct lengths of the struts needed, and hence manufacture and distribute them to the installation crews without spending valuable time making measurements. The rectangular frame structure has the dimensions 4.015.010.0m (height, width, and depth). One sector has a pipe entering the lower comer of the standard frame unit and exiting at the diametrically opposed comer (the one farthest away at the top); call this L1. A second pipe enters and exits at the two different opposite lower comers; call this L2 (Figure 2.74). Figure 2.74 Two pipes cross through a standard frame unit. Write down the vectors along the lines representing those pipes, find the cross product between them from which to create the unit vector n, define a vector that spans two points on each line, and finally determine the minimum distance between the lines. (Take the origin to be at the lower cooler of the ?rst pipe.) Similarly, you may aim develop the symmetric equations for each line and substitute directly into your formula.In the following exercises, points P and Q are given. Let L be the line passing through points P and Q . Find the vector equation of Line L . Find parametric equations of line L . Find symmetric equations of line L . Find parametric equations of the line segment determined by P and Q . 243. P(3,5,9),Q(4,7,2)In the following exercises, points P and Q are given. Let L be the line passing through points P and Q . Find the vector equation of Line L . Find parametric equations of line L . Find symmetric equations of line L . Find parametric equations of the line segment determined by P and Q . 244. P(4,0,5),Q(2,3,1)In the following exercises, points P and Q are given. Let L be the line passing through points P and Q . Find the vector equation of Line L . Find parametric equations of line L . Find symmetric equations of line L . Find parametric equations of the line segment determined by P and Q . 245. P(1,0,5),Q(4,0,3)In the following exercises, points P and Q are given. Let L be the line passing through points P and Q . Find the vector equation of Line L . Find parametric equations of line L . Find symmetric equations of line L . Find parametric equations of the line segment determined by P and Q . 246. P(7,2,6),Q(3,0,6)For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v . Find parametric equations of line L . Find symmetric equations of line L . Find the intersection of the line with the xy-plane. 247. P(1,2,3),v=1,2,3 For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v . Find parametric equations of line L . Find symmetric equations of line L . Find the intersection of the line with the xy-plane. 248. P(3,1,5),v=1,1,1 For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v . Find parametric equations of line L . Find symmetric equations of line L . Find the intersection of the line with the xy-plane. 249. P(3,1,5),v=QR, where Q(2,2,3) and R(3,2,3)For the following exercises, point P and vector v are given. Let L be the line passing through point P with direction v . Find parametric equations of line L . Find symmetric equations of line L . Find the intersection of the line with the xy-plane. 250. P(2,3,0),v=QR, where Q(0,4,5) and R(0,4,6)For the following exercises, line L is given. Find point P that belongs to the line and direction vector v of the line. Express v in component form. Find the distance from the origin to line L . 251. x=1+t,y=3+t,z=5+4t,t For the following exercises, line L is given. Find point P that belongs to the line and direction vector v of the line. Express v in component form. Find the distance from the origin to line L . 252. x=y+1,z=2 For the following exercises, line L is given. Find point P that belongs to the line and direction vector v of the line. Express v in component form. Find the distance from the origin to line L . 253. Find the distance between point A(3,1,1) and the line of symmetric equations x=y=z.For the following exercises, line L is given. Find point P that belongs to the line and direction vector v of the line. Express v in component form. Find the distance from the origin to line L . 254. Find the distance between point A(4,2,5) and the line of parametric equations x=1t,y=t,z=2,t.For the following exercises, lines L1 and L2 are given. Verify whether lines L1 and L2 are parallel. If the lines L1 and L2 are parallel, then find the distance between them. 255. L1:x=1+t,z=2+t,t,L2:x3=y1=z3 For the following exercises, lines L1 and L2 are given. Verify whether lines L1 and L2 are parallel. If the lines L1 and L2 are parallel, then find the distance between them. 256. L1:x=2,y=1,z=t,L2:x=1,y=1,z=23t,t For the following exercises, lines L1 and L2 are given. Verify whether lines L1 and L2 are parallel. If the lines L1 and L2 are parallel, then find the distance between them. 257. Show that the line passing through points P(3,1,0) and Q(1,4,3) is perpendicular to the line with equation x=3t,y=3+8t,z=7+6t,t.For the following exercises, lines L1 and L2 are given. Verify whether lines L1 and L2 are parallel. If the lines L1 and L2 are parallel, then find the distance between them. 258. Are the lines of equations x=2+2t,y=6,z=2+6t and x=1+t,y=1+t,z=t,t, perpendicular to each other?For the following exercises, lines L1 and L2 are given. Verify whether lines L1 and L2 are parallel. If the lines L1 and L2 are parallel, then find the distance between them. 259. Find the point of intersection of the Lines 0f equations x=2y=3z and x=5t,y=1+t,z=t11,t.For the following exercises, lines L1 and L2 are given. Verify whether lines L1 and L2 are parallel. If the lines L1 and L2 are parallel, then find the distance between them. 260. Find the intersection point If the x -axis with the line of parametric equation x=10+t,y=22t,z=3+3t,t.For the following exercises, Lines L1 and L2 are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. 261. L1:x=y1=z and L2:x2=y=z2 For the following exercises, Lines L1 and L2 are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. 262. L1:x=2t,y=0,z=3,t and L2:x=0,y=8+s,z=7+s,s For the following exercises, Lines L1 and L2 are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. 263. L1:x=1+2t,y=1+3t,z=7t,t and L2:x1=23(y4)=27z2 For the following exercises, Lines L1 and L2 are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. 264. L1:3x=y+1=2z and L2:x=6+2t,y=17+6t,z=9+3t,t For the following exercises, Lines L1 and L2 are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. 265. Consider line L of symmetric equations x2=y=z2 and point A(1,1,1). Find parametric equations for a line parallel to L that passes through point A. Find symmetric equations of a line skew to L and that passes through point A. Find symmetric equations of a Line that intersects L and passes through point A.For the following exercises, Lines L1 and L2 are given. Determine whether the lines are equal, parallel but not equal, skew, or intersecting. 266. Consider line L of parametric equations x=t,y=2t,z=3,t. Find parametric equations for a line parallel to L that passes through the origin. Find parametric equations of a line skew to L that passes through the origin. Find symmetric equations 0f a line that intersects L and passes through the origin.For the following exercises, point P and vector n are given. Find the scalar equation of the plane that passes through P and has normal vector n . Find the general form of the equation of the plane that passes through P and has normal vector n . 267. P(0,0,0),n=3i2j+4k For the following exercises, point P and vector n are given. Find the scalar equation of the plane that passes through P and has normal vector n . Find the general form of the equation of the plane that passes through P and has normal vector n . 268. P(3,2,2),n=2i+3j4k For the following exercises, point P and vector n are given. Find the scalar equation of the plane that passes through P and has normal vector n . Find the general form of the equation of the plane that passes through P and has normal vector n . 269. P(1,2,3),n=1,2,3 For the following exercises, point P and vector n are given. Find the scalar equation of the plane that passes through P and has normal vector n . Find the general form of the equation of the plane that passes through P and has normal vector n . 270. P(0,0,0),n=3,2,1 For the following exercises, the equation of a plane is given. Find normal vector n to the plane. Express nusing standard unit vectors. Find the intersections of the plane with the axes of coordinates. Sketch the plane. 271. [T]4x+5y+10z20=0 For the following exercises, the equation of a plane is given. Find normal vector n to the plane. Express nusing standard unit vectors. Find the intersections of the plane with the axes of coordinates. Sketch the plane. 272. 3x+4y12=0 For the following exercises, the equation of a plane is given. Find normal vector n to the plane. Express nusing standard unit vectors. Find the intersections of the plane with the axes of coordinates. Sketch the plane. 273. 3x2y+4z=0 For the following exercises, the equation of a plane is given. Find normal vector n to the plane. Express nusing standard unit vectors. Find the intersections of the plane with the axes of coordinates. Sketch the plane. 274. x+z=0 Given paint P(1,2,3) and vector n=i+j , find point Q on the xaxis such that PQ and n are orthogonal.Show there is no plane perpendicular to n=i+j that passes through points P(1,2,3) and Q(2,3,4) .Find parametric equations 0f the Line passing through point P(2,1,3) that is perpendicular t0 the plane of equation 2x3y+z=7 .Find symmetric equations of the line passing through point P(2,5,4) that is perpendicular t0 the plane of equation 2x+3y5z=0 .Show that line x12=y+13=z24 is parallel to plane x2y+z=6 .Find the real number such that the line of parametric equations x=t,y=2t,z=3+t,t is parallel to the plane of equation x+5y+z10=0.For the following exercises, points P,Q, and R are given. Find the general equation of the plane passing through P,Q, and R . Write the vector equation nPS=0 of the plane at a., where S(x,y,z) is an arbitrary point of the plane. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through P,Q, and R . 281. P(1,1,1),Q(2,4,3), and R(1,2,1)For the following exercises, points P,Q, and R are given. Find the general equation of the plane passing through P,Q, and R . Write the vector equation nPS=0 of the plane at a., where S(x,y,z) is an arbitrary point of the plane. Find parametric equations of the line passing through the origin that is perpendicular to the plane passing through P,Q, and R . 282. P(2,1,4),Q(3,1,3), and R(2,1,0)Consider the planes of equations x+y+z=1 and x+z=0 . a. Show that the planes intersect. b. Find symmetric equations of the line passing through point P(1,4,6) that is parallel to the line of intersection of the planes.Consider the planes of equations y+z2=0 and xy=0 . Show that the planes intersect. Find parametric equations of the line passing through point P(8,0,2) that is parallel to the line of intersection 0f the planes.Find the scalar equation of the plane that passes through paint P(1,2,1) and is perpendicular to the line of intersection 0f planes x+yz2=0 and 2xy+3z1=0 .Find the general equation of the plane that passes though the origin and is perpendicular to the line of intersection of planes x+y+2=0 and z3=0 .Determine whether the Line of parametric equations x=1+2t,y=2t,z=2+t,t intersects the plane with equation 3x+4y+6z7=0 . If it does intersect, find the point of intersection.Determine whether the Line of parametric equations x=5,y=4t,z=2t,t intersects the plane with equation 2xy+z=5. If it does intersect, find the point of intersection.Find the distance from point P(1,5,4) to the plane of equation 3xy+2z6=0 .Find the distance from point P(1,2,3) to the plane of equation (x3)+2(y+1)4z=0 .For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. 291. [T]x+y+z=0,2xy+z7=0 For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. 292. 5x3y+z=4,x+4y+7z=1 For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. 293. x5yz=1,5x25y5z=3 For the following exercises, the equations of two planes are given. Determine whether the planes are parallel, orthogonal, or neither. If the planes are neither parallel nor orthogonal, then find the measure of the angle between the planes. Express the answer in degrees rounded to the nearest integer. 294. [T]x3y+6z=4,5x+yz=4 Show that the lines of equations x=t,y=1+t,z=2+t,t and x2=y13=z3 are skew, and find the distance between them.Show that the lines of equations x=1+t,y=2+t,z=3t,t, and x=5+s,y=8+2s,z=7s,s, are skew, and find the distance between them.Consider point C(3,2,4) and the plane of equation 2x+4y3z=8. Find the radius of the sphere with center Ctangent to the given plane. Find point P of tangency.Consider the plane of equation xyz8=0. Find the equation of the sphere with center C at the origin that is tangent to the given plane. Find parametric equations of the line passing through the origin and the point of tangency.Two children are playing with a ball. The girl throws the ball to the boy. The ball travels in the air, curves 3ft to the right, and falls 5ft away from the girl (see the following figure). If the plane that contains the trajectory of the ball 15 perpendicular to the ground, find its equation.[T] John allocates d dollars to consume monthly three goods of prices a,b, and c. In this context, the budget equation is de?ned as ax+by+cz=d, where x0,y0, and z0 represent the number of items bought from each of the goods. The budget set is given by {(x,y,z)|ax+by+czd,x0,y0,}, and the budget plane is the part of the plane of equation ax+by+cz=d for which x0,y0, and z0. Consider a=8,b=5,c=10, and d=500. Use a CAB to graph the budget set and budget plane. For z=25, find the new budget equation and graph the budget set ill the same system of coordinates.[T] Consider r(t)=sint,cost,2t the position vector of a particle at time t[0,3], where the components of r are expressed in centimeters and time is measured in seconds. Let OP be the position vector of the particle after 1sec. Determine the velocity vector v(1) of the particle after 1sec. Find the scalar equation of the plane that is perpendicular to v(1) and passes through point P. This plane is called the normal plane to the path of the particle at point P. Use a CAS to visualize the path of the particle along with the velocity vector and normal plane at point P.[T] A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) A(8,0,0),B(8,18,0),C(0,18,8), and D(0,0,8) (see the following figure). Find the general form of the equation of the plane that contains the solar panel by using points A,B, and C, and show that its normal vector is equivalent to ABAD. Find parametric equations of line L1 that passes through the center of the solar panel and has direction vector s=13i+13j+13k, which points toward the position of the Sun at a particular time of day. equations of line L2 that passes through the center of the solar panel and is perpendicular to it. Determine the angle of elevation of file Sun above the solar panel by using the angle between lines L1 and L2.For the following exercises, sketch and describe the cylindrical surface of the given equation. 303. [T]x2+z2=1 For the following exercises, sketch and describe the cylindrical surface of the given equation. 304. [T]x2+y2=9 For the following exercises, sketch and describe the cylindrical surface of the given equation. 305. [T]z=cos(2+x)For the following exercises, sketch and describe the cylindrical surface of the given equation. 306. [T]z=ex For the following exercises, sketch and describe the cylindrical surface of the given equation. 307. [T]z=9y2 For the following exercises, sketch and describe the cylindrical surface of the given equation. 308. [T]z=ln(x)For the following exercises, the graph of a quadric surface is given. Specify the name of the quadric surface. Determine the axis of symmetry of the quadric surface. 309.For the following exercises, the graph of a quadric surface is given. Specify the name of the quadric surface. Determine the axis of symmetry of the quadric surface. 310.For the following exercises, the graph of a quadric surface is given. Specify the name of the quadric surface. Determine the axis of symmetry of the quadric surface. 311.For the following exercises, the graph of a quadric surface is given. Specify the name of the quadric surface. Determine the axis of symmetry of the quadric surface. 312.For the fallowing exercises, match the given quadric surface with its corresponding equation in standard form. x24+y29z212=1x24y29z212=1x24+y29+z212=1z2=4x2+3y2z=4x2y24x2+y2z2=0 313. Hyperboloid of two sheets For the fallowing exercises, match the given quadric surface with its corresponding equation in standard form. x24+y29z212=1x24y29z212=1x24+y29+z212=1z2=4x2+3y2z=4x2y24x2+y2z2=0 314. Ellipsoid For the fallowing exercises, match the given quadric surface with its corresponding equation in standard form. x24+y29z212=1x24y29z212=1x24+y29+z212=1z2=4x2+3y2z=4x2y24x2+y2z2=0 315. Elliptic paraboloid For the fallowing exercises, match the given quadric surface with its corresponding equation in standard form. x24+y29z212=1x24y29z212=1x24+y29+z212=1z2=4x2+3y2z=4x2y24x2+y2z2=0 316. Hyperbolic paraboloid For the fallowing exercises, match the given quadric surface with its corresponding equation in standard form. x24+y29z212=1x24y29z212=1x24+y29+z212=1z2=4x2+3y2z=4x2y24x2+y2z2=0 317. Hyperboloid of one sheet For the fallowing exercises, match the given quadric surface with its corresponding equation in standard form. x24+y29z212=1x24y29z212=1x24+y29+z212=1z2=4x2+3y2z=4x2y24x2+y2z2=0 318. Elliptic cone For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 319. x2+36y2+36z2=9 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 320. 4x2+25y2+z2=100 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 321. 3x2+5y2z2=10 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 322. 3x2y26z2=18 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 323. 5y=x2z2 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 324. 8x25y210z=0 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 325. x2+5y2+3z215=0 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 326. 63x2+7y2+9z263=0 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 327. x2+5y28z2=0 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 328. 5x24y2+20z2=0 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 329. 6x=3y2+2z2 For the following exercises, rewrite the given equation of the quadric surface in standard form. Identify the surface. 330. 49y=x2+7z2 For the following exercises, find the trace of the given quadric surface in the specifled plane of coordinates and sketch it. 331. [T]x2+z2+4y=0,z=0 For the following exercises, find the trace of the given quadric surface in the specifled plane of coordinates and sketch it. 332. [T]x2+z2+4y=0,x=0 For the following exercises, find the trace of the given quadric surface in the speci?ed plane of coordinates and sketch it. 333. [T]4x2+25x2+z2=100,x=0 For the following exercises, find the trace of the given quadric surface in the specifled plane of coordinates and sketch it. 334. [T]4x2+25y2+z2=100,y=0 For the following exercises, find the trace of the given quadric surface in the specifled plane of coordinates and sketch it. 335. [T]x2+y24+z2100=1,x=0 For the following exercises, find the trace of the given quadric surface in the specifled plane of coordinates and sketch it. 336. [T]x2yz2=1,y=0 Use the graph of the given quadric surface to answer the questions. Specify the name of the quadric surface. Which of the equations— 16x2+9y2+36z2=3600,9x2+36y2+16z2=3600, or 36x2+9y2+16z2=3600—corresponds to the graph? Use b. to write the equation of the quadric surface in standard form.Use the graph of the given quadric surface to answer the questions. Specify the name of the quadric surface. Which of the equations 36z=9x2+y2,9x2+4y2=36z, or 36z=81x2+4y2—correspunds to the graph above? Use b. to write the equation of the quadric surface in standard form.For the following exercises, the equation of a quadric surface is given. Use the method of completing the square to write the equation in standard form. Identify the surface. 339. x2+2z2+6x8z+1=0 For the following exercises, the equation of a quadric surface is given. Use the method of completing the square to write the equation in standard form. Identify the surface. 340. 4x2y2+z28x+2y+2z+3=0 For the following exercises, the equation of a quadric surface is given. Use the method of completing the square to write the equation in standard form. Identify the surface. 341. x2+4y24z26x16y16z+5=0 For the following exercises, the equation of a quadric surface is given. Use the method of completing the square to write the equation in standard form. Identify the surface. 342. x2+z24y+4=0 For the following exercises, the equation of a quadric surface is given. Use the method of completing the square to write the equation in standard form. Identify the surface. 343. x2+y24z23+6x+9=0 For the following exercises, the equation of a quadric surface is given. Use the method of completing the square to write the equation in standard form. Identify the surface. 344. x2y2+z212z+2x+37=0 Write the standard form of the equation of the ellipsoid centered at the origin that passes through points A(2,0,2),B(0,0,1), and C(12,11,12).Write the standard form of the equation of the ellipsoid neutered at point P(1,1,0) that passes through points A(6,1,0),B(4,2,0) and C(1,2,1).Determine the intersection points of elliptic cone x2y2z2=0 with the line of symmetric equations x12=y+13=z.Determine the intersection points of parabolic hyperboloid z=3x22y2 with the line of parametric equations x=3t,y=2t,z=19t, where t.Find the equation of the quadric surface with points P(x,y,z) that are equidistant from point Q(0,1,0) and plane of equation y=1. Identify the surface.Find the equation of the quadric surface with paints P(x,y,z) that are equidistant from point Q(0,2,0) and plane of equation y=2. Identify the surface.If the surface of a parabolic reflector is described by equation 400z=x2+y2 , find the focal point of the reflector.Consider the parabolic reflector described by equation z=20x2+20y2 . Find its focal point.Show that quadric surface x2+y2+z2+2xy+2xz+2yz+x+y+z=0 reduces to two parallel planes.Show that quadric surface x2+y2+z22xy+2yz1=0 reduces to two parallel planes passing.[T] The intersection between cylinder (x1)2+y2=1 and sphere x2+y2+z2=4 is called a Viviani curve. Solve the system consisting of the equations of the surfaces to find the equation of the intersection curve. (Hint: Find x and y in terms of z .) Use a computer algebra system (GAS) to Visualize the intersection curve on sphere x2+y2+z2=4 .Hyperboloid of one sheet 25x2+25y2z2=25 and elliptic cone 25x2+75y2+z2=0 are represented in the following figure along with their intersection curves. Identify the intersection curves and find their equations (Hint: Find y from the system consisting of the equations of the surfaces.)[T] Use a CA8 to create the intersection between cylinder 9x2+4y2=18 and ellipsoid 36x2+16y2+9z2=144 , and find the equations of the intersection curves.[T] A spheroid is an ellipsoid with two equal semiaxes. For instance, the equation of a spheroid with the z-axis as its axis of symmetry is given by x2a2+y2a2+z2c2=1, where a and c are positive real numbers. The spheroid is called oblate if ca, and prolate for ca. The eye cornea is approximated as a prolate spheroid with an axis that is the eye, where a=8.7mm and c=9.6mm. Write the equation of the spheroid that models the cornea and sketch the surface. Give two examples of objects with predate spheroid shapes.[T] In cartography, Earth is approximated by all oblate spheroid rather than a sphere. The radii at the equator and poles are approximately 3963 mi and 3950 mi, respectively. a. Write the equation in standard form of the ellipsoid that represents the shape of Earth. Assume the center of Earth is at the origin and that the trace formed by plane z=0 corresponds to the equator. b. Sketch the graph. c. Find the equation of the intersection curve of the surface with plane z=1000 that is parallel to the xy -plane. The intersection curve is called a parallel. d. Find the equation of the intersection curve of the surface with plane x+y=0 that passes through the z-axis. The intersection curve is called a meridian.[T] A set 0f buzzing stunt magnets (or “rattlesnake eggs”) includes two sparkling, polished, super strong spheroid-shaped magnets well-known for children’s entertainment. Each magnet is 1.625 in. long and 0.5 in. wide at the middle. While tossing them jute the air, they create a buzzing sound as they attract each other. Write the equation of the prolate spheroid centered at the origin that describes the shape of one of the magnets. Write the equations 0f the prolate spheroids that model the shape of the buzzing stunt magnets. Use a CAS to create the graphs.[T] A heart-shaped surface is given by equation (x2+94y2+z21)3x2z3980y2z3=0. a. Use a CAS to graph the surface that models this shape. b. Determine and sketch the trace 0f the heart-shaped surface on the xz -plane.[T] The ring torus symmetric about the z -axis is a special type of surface in topology and its equation is given by (x2+y2+z2+R2r2)=4R2(x2+y2), where Rr0 . The numbers R and r are called are the major and minor radii, respectively, of the surface. The following figure shows a ring torus for which R=2 and r=1. Write the equation of the ring torus with R=2 and r=1, and use a CAS to graph the surface. Compare the graph with the ?gure given. Determine the equation and sketch the trace of the ring torus from a. on the xy-plane. Give two examples of objects with ring torus shapes.Use the following ?gure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates (r,,z) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 363. (4,6,3)Use the following ?gure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates (r,,z) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 364. (3,3,5)Use the following ?gure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates (r,,z) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 365. (4,76,3)Use the following ?gure as an aid in identifying the relationship between the rectangular, cylindrical, and spherical coordinate systems. For the following exercises, the cylindrical coordinates (r,,z) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 366. (2,,4)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the cylindrical coordinates (r,,z) of the point. 367. (1,3,2)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the cylindrical coordinates (r,,z) of the point. 368. (1,1,5)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the cylindrical coordinates (r,,z) of the point. 369. (3,3,7)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the cylindrical coordinates (r,,z) of the point. 370. (22,22,4)For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 371. [T]r=4 For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 372. [T]z=r2cos2 For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 373. [T]r2cos(2)+z2+1=0 For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 374. [T]r=3sin For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 375. [T]r=2cos For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 376. [T]r2+z2=5 For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 377. [T]r=2sec For the following exercises, the equation of a surface in cylindrical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 378. [T]r=3csc For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. 379. z=3 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. 380. x=6 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. 381. x2+y2+z2=9 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. 382. y=2x2 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. 383. x2+y216x=0 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in cylindrical coordinates. 384. x2+y23x2+y2+2=0 For the following exercises, the spherical coordinates (,,) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 385. (3,0,)For the following exercises, the spherical coordinates (,,) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 386. (1,6,6)For the following exercises, the spherical coordinates (,,) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 387. (12,4,4)For the following exercises, the spherical coordinates (,,) of a point are given. Find the rectangular coordinates (x,y,z) of the point. 388. (3,4,6)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the spherical coordinates (,,) of the paint. Express the measure of the angles in degrees rounded to the meanest integer. 389. (4,0,0)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the spherical coordinates (,,) of the paint. Express the measure of the angles in degrees rounded to the meanest integer. 390. (1,2,1)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the spherical coordinates (,,) of the paint. Express the measure of the angles in degrees rounded to the meanest integer. 391. (0,3,0)For the following exercises, the rectangular coordinates (x,y,z) of a point are given. Find the spherical coordinates (,,) of the paint. Express the measure of the angles in degrees rounded to the meanest integer. 392. (2,23.4)For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 393. [T]=3 For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 394. [T]=3 For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 395. [T]=2cos For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 396. [T]=4csc For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 397. [T]=2 For the following exercises, the equation of a surface in spherical coordinates is given. Find the equation of the surface in rectangular coordinates. Identify and graph the surface. 398. [T]=6cscsec For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface. 399. x2+y23z2=0,z0 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface. 400. x2+y2+z24z=0 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface. 401. z=6 For the following exercises, the equation of a surface in rectangular coordinates is given. Find the equation of the surface in spherical coordinates. Identify the surface. 402. x2+y2=9 For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle in radians rounded to four decimal places. 403. [T](1,4,3)For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle in radians rounded to four decimal places. 404. [T](5,,12)For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle in radians rounded to four decimal places. 405. (3,2,3)For the following exercises, the cylindrical coordinates of a point are given. Find its associated spherical coordinates, with the measure of the angle in radians rounded to four decimal places. 406. (3,6,3)For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. 407. (2,4,2)]For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. 408. (4,4,6)For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. 409. (8,3,2)For the following exercises, the spherical coordinates of a point are given. Find its associated cylindrical coordinates. 410. (9,6,3)For the following exercises, find the most suitable system of coordinates to describe the solids. 411. The solid situated in the first octant with a vertex at the origin and enclosed by a cube of edge length a, where a0 For the following exercises, find the most suitable system of coordinates to describe the solids. 412. A spherical shell determined by the region between two concentric spheres centered at the origin, of radii of a and b, respectively, where ba0 For the following exercises, find the most suitable system of coordinates to describe the solids. 413. A solid inside sphere x2+y2+z2=9 and outside cylinder (x32)2+y2=94 For the following exercises, find the most suitable system of coordinates to describe the solids. 414. A cylindrical shell of height 10 determined by the region between two cylinders with the same center, parallel rulings, and radii of 2 and 5, respectively For the following exercises, find the most suitable system of coordinates to describe the solids. 415. [T] Use a CAS to graph in cylindrical coordinates the region between elliptic paraboloid z=x2+y2 and cone x2+y2z2=0.For the following exercises, find the most suitable system of coordinates to describe the solids. 416. [T] Use a CAB to graph in spherical coordinates the “ice cream-coue region” situated above the xy -plane between sphere x2+y2+z2=4 and elliptical cone x2+y2z2=0.For the following exercises, find the most suitable system of coordinates to describe the solids. 417. Washington, DC, is located at 39 N and 77 W (see the following figure). Assume the radius of Earth is 4000mi. Express the location of Washington, DC, in spherical coordinates.For the following exercises, find the most suitable system of coordinates to describe the solids. 418. San Francisco is located at 37.78N and 122.42W . Assume the radius of Earth is 4000mi. Express the location of San Francisco in spherical coordinates.For the following exercises, find the most suitable system of coordinates to describe the solids. 419. Find the latitude and longitude of Rio de Janeiro if its spherical coordinates are (4000,43.17,102.91).For the following exercises, find the most suitable system of coordinates to describe the solids. 420. Find the latitude and longitude of Berlin if its spherical coordinates are (4000,13.38,37.48)For the following exercises, find the most suitable system of coordinates to describe the solids. 421. [T] Consider the torus of equation (x2+y2+z2+R2r2)=4R2(x2+y2), where Rr0. a. Write the equation of the torus in spherical coordinates. b. If R=r , the surface is called a horn torus. Show that the equation 0f a horn torus in spherical coordinates is =2Rsin. c. Use a CAB to graph the horn torus with R=r=2 in spherical coordinates.For the following exercises, find the most suitable system of coordinates to describe the solids. 422. [T] The “bumpy sphere” with an equation in spherical coordinates is =a+bcos(m)sin(n), a with [0,2] and go [0,], where a and b are positive numbers and m and n are positive integers, may be used in applied mathematics to model tumor growth. Show that the “bumpy sphere” is contained inside a sphere of equation , =a+b . Find the values of and at which the two surfaces intersect. Use 3 CAS to graph the surface for a=14,b=2,m=4, and n=6 slung with sphere =a+b. Find the equation of the intersection curve of the surface at b. with the cane =12. Graph the intersection curve in the plane of intersection.For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample. 423. For vectors a and b and any given scalar c , c(ab)=(ca)b .For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample. 424. For vectors a and b and any given scalar c , c(ab)=(ca)b .For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample. 425. The symmetric equation for the line of intersection between two planes x+y+z=2 and x+2y4z=5 is given by x16=y15=z .For the following exercises, determine whether the statement is true or false. Justify the answer with a proof or a counterexample. 426. If ab=0, then a is perpendicular to b .For the following exercises, use the given vectors to fill the quantities. 427. a=9i2j,b=3i+j 3a+b00 |a| a|b|a b|a For the following exercises, use the given vectors to fill the quantities. 428. a=2i+j9k,b=i+2k,c=4i2j+k 2 ab |bc| b|bc| c|ba|For the following exercises, use the given vectors to fill the quantities. 429. Find the values 0f a such that vectors 2,4,a and 0,1,a are orthogonal.For the following exercises, find the unit vectors. 430. Find the unit vector that has the same direction as vector v that begins at (0,3) and ends at (4,10).For the following exercises, find the unit vectors. 431. Find the unit vector that has the same direction as vector v that begins at (1,4,10) and ends at (3,0,4).For the following exercises, find the area or volume of the given shapes. 432. The parallelogram spanned by vectors a=1,13 and b=3,21 For the following exercises, find the area or volume of the given shapes. 433. The parallelepiped formed by a=1,4,1 and b=3,6,2, and c=2,1,5 .For the following exercises, find the vector and parametric equations of the line with the given properties. 434. The line that passes through point (2,3,7) that is parallel to vector 1,3,2 .For the following exercises, find the vector and parametric equations of the line with the given properties. 435. The line that passes through points (1,3,5) and (2,6,3) .For the following exercises, find the equation of the plane with the given properties. 436. The plane that passes through point (4,7,1) and has normal vector n=3,4,2 .For the following exercises, find the equation of the plane with the given properties. 437. The plane that passes through paints (0,1,5),(2,1,6), and (3,2,5) .For the following exercises, find the traces for the surfaces in planes x=k,y=k, and z=k . Then, describe and draw the surfaces. 438. 9x2+4y216y+36z2=20 For the following exercises, find the traces for the surfaces in planes x=k,y=k, and z=k . Then, describe and draw the surfaces. 439. x2=y2+z2 For the fallowing exercises, write the given equation in cylindrical coordinates and spherical coordinates. 440. x2+y2+z2=144 For the fallowing exercises, write the given equation in cylindrical coordinates and spherical coordinates. 441. z=x2+y21 For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface. 442. 2(sin2()cos2()=1)For the following exercises, convert the given equations from cylindrical or spherical coordinates to rectangular coordinates. Identify the given surface. 443. r22rcos()+z2=1 For the following exercises, consider a small boat crossing a river. 444. If the boat velocity is 5km/h due north in still water and the water has a current of 2km/h due west (see the following ?gure), what is the velocity of the boat relative to shore? What is the angle that the boat is actually traveling?For the following exercises, consider a small boat crossing a river. 445. When the boat reaches the shore, two ropes are thrown to people to help pull the boat ashore. One rope is at an angle of 25 and the other is at 35 . If the boat must be pulled straight and at a force of 500N , find the magnitude of force for each rope (see the following figure}.An airplane is flying in the direction of 52 east of north with a speed of 450mph . A strong wind has a hearing 33 east of north with a speed of 50mph . What is the resultant ground speed and bearing of the airplane?Calculate the work done by moving a particle from position (1,2,0) to (8,4,5) along a straight line with a force F=2i+3jk .The following problems consider your unsuccessful attempt to take the lire off your car using a wrench to loosen the bolts. Assume the wrench is 0.3m long and you are able to apply a 200-N force. 448. Because your lire is flat, you are only able to apply your force at a 60 angle. What is the torque at the center of the bolt? Assume this force is not enough to loosen the bolt.The following problems consider your unsuccessful attempt to take the lire off your car using a wrench to loosen the bolts. Assume the wrench is 0.3m long and you are able to apply a 200-N force. 449. Someone lends you a tire jack and you are now able to apply a 200-N force at an 80 angle. Is your resulting torque going to be more or less? What is the new resulting torque at the center of the bolt? Assume this force is not enough to loosen the bolt.Give the component functions x=f(t) and y=g(t) for the vector-valued function r(t)=3secti+2tantj.Given r(t)=3secti+2tantj , find the following values (if possible). r(4) r() r(2)Sketch the curve of the vector-valued function r(t)=3secti+2tantj and give the orientation of the curve. Sketch asymptotes as a guide to the graph.Evaluate limt0eti+sinttj+etk .Given the vector-valued function r(t)=cost,sint , find the following values: limt4r(t) r(3) Is r(t) continuous at t=3? Graph r(t) .Given the vector-valued function r(t)=t,t2+1 , find the following values: limt3r(t) r(3) Is r(t) continuous at x=3? r(t+2)r(t)Let r(t)=eti+sintj+lntk . Find the following values: r(4) limt/4r(t) Is r(t) continuous at t=t=4?Find the limit of the following vector-valued functions at the indicated value of t. 8. limt4t3,t2t4,tan(t)Find the limit of the following vector-valued functions at the indicated value of t. 9. limt/2r(t)forr(t)=eti+sintj+lntk Find the limit of the following vector-valued functions at the indicated value of t. 10. limte2t,2t+33t1,arctan(2t)Find the limit of the following vector-valued functions at the indicated value of t. 11. limte2tln(t),lntt2,ln( t 2)Find the limit of the following vector-valued functions at the indicated value of t. 12. limt/6cos2t,sin2t,1 Find the limit of the following vector-valued functions at the indicated value of t. 13. limtr(t)forr(t)=2eti+etj+ln(tl)k Find the limit of the following vector-valued functions at the indicated value of t. 14. Describe the curve defined by the vector-valued function r(t)=(1+t)i+(2+5t)j+(1+6t)k .Find the domain of the vector-valued functions. 15. Domain: r(t)=t2,tant,lnt Find the domain of the vector-valued functions. 16. Domain: r(t)=t2,t3,32t+1 Find the domain of the vector-valued functions. 17. Domain: r(t)=csc(t),1t3,ln(t2)Let r(t)=cost,t,sint and use it to answer the following questions. 18. For what values of t is r(t) continuous?Let r(t)=cost,t,sint and use it to answer the following questions. 19. Sketch the graph of r(t) .Let r(t)=cost,t,sint and use it to answer the following questions. 20. Find the domain of r(t)=2eti+etj+ln(t1)k .Let r(t)=cost,t,sint and use it to answer the following questions. 21. For what values of r is r(t)=2eti+etj+ln(t1)k continuous?Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. 22. r(t)=2ti+t2j (Hint: Let x=2t and y=t2 . Solve the first equation for x in terms of t and substitute this result into the second equation.)Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. 23. r(t)=t3i+2tj Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. 24. r(t)=2(sinht)i+2(cosht)j,t0 Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. 25. r(t)=3(cost)i+3(sint)j Eliminate the parameter t, write the equation in Cartesian coordinates, then sketch the graphs of the vector-valued functions. 26. r(t)=3sint,3cost Use a graphing utility to sketch each of the following vector-valued functions: 27. [T]r(t)=2cost2i+(2t)j Use a graphing utility to sketch each of the following vector-valued functions: 28. [T]r(t)=ecos(3t),esin(t)Use a graphing utility to sketch each of the following vector-valued functions: 29. [T]r(t)=2sin(2t),3+2cost Use a graphing utility to sketch each of the following vector-valued functions: 30. 4x2+9y2=36 ; clockwise and counterclockwise Use a graphing utility to sketch each of the following vector-valued functions: 31. r(t)=t,t2 ; from left to right Use a graphing utility to sketch each of the following vector-valued functions: 32. The line through P and Q where P is (1,4,2) and Q is (3,9,6)Use a graphing utility to sketch each of the following vector-valued functions: Consider the curve described by the vector—valued function r(t)=(50etcost)i+(50etsint)j+(55et)k 33. What is the initial point of the path corresponding to r(0) ?Use a graphing utility to sketch each of the following vector-valued functions: Consider the curve described by the vector—valued function r(t)=(50etcost)i+(50etsint)j+(55et)k 34. What is limtr(t) ?Use a graphing utility to sketch each of the following vector-valued functions: Consider the curve described by the vector—valued function r(t)=(50etcost)i+(50etsint)j+(55et)k 35. [T] Use technology to sketch the curve.Use a graphing utility to sketch each of the following vector-valued functions: Consider the curve described by the vector—valued function r(t)=(50etcost)i+(50etsint)j+(55et)k 36. Eliminate the parameter t to show that z=5r10 where r2=x2+y2 .[T] Let r(t)=costi+sintj+0.3sin(2t)k . Use technology to graph the curve (called the roller-coaster curve) over the interval [0,2) . Choose at least two views to determine the peaks and valleys.[T] Use the result of the preceding problem to construct an equation of a roller coster with a steep drop from the peak and steep incline from the “valley.” Then, use technology to graph the equation.Use the results If the preceding two problems to construct an equation of a path of a roller coster with more than two turning points (peaks and valleys).a. Graph the curve r(t)=(4+cos(18t))cos(t)i+(4+cos(18t)sin(t))j+0.3sin(18t)k ; using two viewing angles of your choice to see the overall shape of the curve. b. Does the curve resemble a “slinky”? c. What changes to the equation should be made to increase the number of coils of the slinky?Compute the derivatives of the vector-valued functions. 41. r(t)=t3i+3t2j+t36k Compute the derivatives of the vector-valued functions. 42. r(t)=sin(t)i+cos(t)j+etk Compute the derivatives of the vector-valued functions. 43. r(t)=eti+sin(3t)j+10tk . A sketch of the graphs is shown here. Notice the varying periodic nature of the graph.Compute the derivatives of the vector-valued functions. 44. r(t)=eti+2etj+k Compute the derivatives of the vector-valued functions. 45. r(t)=i+j+k Compute the derivatives of the vector-valued functions. 46. r(t)=teti+tln(t)j+sin(3t)k Compute the derivatives of the vector-valued functions. 47. r(t)=1t+1i+arctan(t)j+lnt3k Compute the derivatives of the vector-valued functions. 48. r(t)=tan(2t)i+sec(2t)j+sin2(t)k Compute the derivatives of the vector-valued functions. 49. r(t)=3i+4sin(3t)j+tcos(t)k Compute the derivatives of the vector-valued functions. 50. r(t)=t2i+te2tj5e4tk For the following problems, find a tangent vector at the indicated value of t. 51. r(t)=ti+sin(2t)j+cos(3t)k;t=3 For the following problems, find a tangent vector at the indicated value of t. 52. r(t)=3t3i+2t2j+1tk;t=1 For the following problems, find a tangent vector at the indicated value of t. 53. r(t)=3eti+2e3tj+4e2tk;t=ln(2)For the following problems, find a tangent vector at the indicated value of t. 54. r(t)=cos(2t)i+2sintj+t2k;t=2 Find the unit tangent vector for the following parameterized curves. 55. r(t)=6i+cos(3t)j+3sin(4t)k,0t2 Find the unit tangent vector for the following parameterized curves. 56. r(t)=costi+sintj+sintk,0t2. Two views of this curve are presented here:Find the unit tangent vector for the following parameterized curves. 57. r(t)=3cos(4t)i+3sin(4t)j+5tk,1t2 Find the unit tangent vector for the following parameterized curves. 58. r(t)=ti+3tj+t2k Let r(t)=ti+t2jt4k and s(t)=sin(t)i+etj+cos(t)k . Here is the graph of the function:Find the following. 59. ddt[r(t2)]Find the following. 60. ddt[t2.s(t)]Find the following. 61. ddt[r(t).s(t)]Compute the first, second, and third derivatives of r(t)=3ti+6ln(t)j+5e3tk .Find r(t)r(t) for r(t)=3t5i+5tj+2t2k .The acceleration function, initial velocity, and initial position of a particle are a(t)=5costi5sintj,v(0)=9i+2j , and r(0)=5i . Find v(t) and r(t) .The position vector of a particle is r(t)=5sec(2t)i4tan(t)j+7t2k. Graph the position function and display a View of the graph that illustrates the asymptotic behavior of the function. Find the velocity as t approaches but is not equal to /4 (if it exists).Find the velocity and the speed of a panicle with the position function r(t)=(2t1)(2t+1)i+ln(14t2)j . The speed of a particle is the magnitude of the velocity and is represented by r(t) . A particle moves on a Circular path of radius b according to the function r(t)=bcos(t)i+bsin(t)j , where is the angular velocity, d/dt .Find the velocity function and show that v(t) is always orthogonal to r(t) .Show that the speed of the particle is proportional to the angular velocity.Evaluate ddt[u(t)u(t)] given u(t)=t2i2tj+k .Find the antiderivative of r(t)=cos(2t)i2sintj+11+t2k that satisfies the initial condition r(0)=3i2j+k .Evaluate 03ti+t2jdt .An object starts from nest at point P(1,2,0) and moves with an acceleration of a(t)=j+2k , where a(t) is measured in feet per second per second. Find the location of the object after t=2sec .Show that if the speed 0f a particle traveling along a curve represented by a vector-valued function is constant, then the velocity function is always perpendicular to the acceleration function.Given r(t)=ti+3tj+t2k and u(t)=4ti+t2j+t3k , find ddt(r(t)u(t)) .Given r(t)=t+cost,tsint , find the velocity and the speed at any time.Find the velocity vector for the function r(t)=et,et,0 .Find the equation of the tangent line to the curve r(t)=et,et,0 at t=0 .Describe and sketch the curve represented by the vector-valued function r(t)=6t,6tt2 .Locate the highest point on the curve r(t)=6t,6tt2 and give the value of the function at this point.The position vector for a particle is r(t)=ti+t2j+t3k . The graph is shown here: 80. Find the velocity vector at any time.

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