Steepest ascent on a plane Suppose a long sloping hillside is described by the plane z = ax + by + c, where a, b, and c are constants. Find the path in the xy-plane, beginning at (x0, y0), that corresponds to the path of steepest ascent on the hillside.

Q: Use Lagrange multipliers to find the highest point on the curve of intersection of the surfaces.…

A: We have given the equation of sphere: x2 + y2 + z2 = 36 and plane: 2x + y − z = 2 We have to find…

Q: 1. A bug crawls on the plane z = 2x +y, from the point (-2, 4, 0) to the point (2, 4, 8) along a…

A: a) Given equation of plane is z=2x+y. Write this equation in the form y=z-2x. The path of bug along…

Q: Cube F = 2xyi + 2yzj + 2xzkD: The cube cut from the first octant by the planes x = 1, y = 1,z = 1

A: GIvencube F=2xyi+2yzj+2xzk Boundary D: x=1,y=1,z=1Tofind The outward flux of F across the boundary…

Q: Midpoint formula Prove that the midpoint of the line segment joining P(x1. y1- Z,) and Q(x,. Y2, Z,)…

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Q: Plane x+4y-3z=1 is perpendicular to plane -3x +6y+7z D0- O True O False

A: normal vector of plane x+4y-3z=1 is <1,4,-3>and normal vector of plane -3x+6y+7z=0 is…

Q: The plane + + = 1 intersects the x-, y-, and z-axes in points P, Q, R. Find the area of the triangle…

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Q: Tangent of x2/3 + y2/3 + z2/3 = a²/3 surface at any point (Xo , Yo , Zo ) Show that the sum of the…

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Q: 3) Suppose plane P, contains the points A(5,0, –1), B(2,3, 2), and C(-2, 1, –4), and assume P, is…

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Q: A cariable straight line drawn through the point of intersection of the straight lines x/a+y/b=1 and…

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Q: The line L: x = 3 + 2t, y = 2t, z = t intersects the plane x + 3y - z =-4 in a point P. Find the…

A: Given:

Q: Find the distance from the point (5, 5, 5) to the line x = 0, y = 5+ t, z = 5+t. Find the point P…

A: Explanation of the answer is as follows

Q: Give an equation of the plane determined by the intersecting lines L1 and L2 L1 : x = 2t y = 4 – t z…

A: The equation of two lines is given as follows: L1: x=2t, y=4-t, z=3+t L2: x=2+s, y=-3+2s, z=s The…

Q: Find equations for all the planes that intersect the y-axis at y =1 and the z-axis at z =2, and are…

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Q: Horizontal plane Q and vertical plane N intersect.

A: Explanation: Here we will draw the two planes Q is horizontal plane & N is vertical plane.

Q: Tangent of x2/3 + y2/3 + z2/3 = a²/3 surface at any point ( xo ,Yo ,Zo ) Show that the sum of the…

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Q: Disk Method y = x^(2)-2x, x-axis ; about the x-axis

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Q: equantion for the tongent plane to the quadratic Z =4x² +y" at the point (4,4,5). Find an surface

A: we have to find the equation for the tangent plane

Q: Tangent of x2/3 + y2/3 + z2/3 = a²/3 surface at any point ( xo , Yo , Zo ) Show that the sum of the…

A: Solution:

Q: the xyz-plane determine the distance and angle from point (2,1,1) in the space and two planes…

A: The distance between point Px1,y1,z1 and the plane Ax+By+Cz+D=0 is  d=Ax1+By1+Cz1+DA2+B2+C2

Q: det äLt)=cost ēit test(2-t-cost) ēga Show that the plane ď is plane curve, and defermine that curve…

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Q: the origin. What are the highest and lowest temperatures encountered by the ant? c) Suppose that the…

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Q: Find an equation of the plane. the plane through the point  (8, −3, −4)  and parallel to the plane…

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Q: nterseeting Lines Finad the plane determined by the Loi Xi=l-2t, , J="l-st,, 2= -2+t, L2: X2=-3+t2,…

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Q: The amazing quadrilateral property—with coordinates Prove the quadrilateral property, assuming the…

A: It is given that the co-ordinates are Px1,y1,0,Qx2,y2,0,Rx3,y3,0 and Sx4,y4,z4

Q: Parabolic dome H(x, y, z) = x2√5- 4z, over the parabolicdome z = 1 - x2 - y2, z ≥ 0

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Q: Find the equetion of the plane contains the line x=3t,y=1+t, z=2t, and parallel to the line of…

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Q: Designing a gondola The designer of a 30-ft-diameter spherical hot air balloon wants to suspend the…

A: Given information:

Q: Show that planes 2x- y + z = 0 and x+2y-z = 1 intersect according to a straight r.

A: General Equation of a Plane: A plane can be represented by an equation of first degree in such as…

Q: n equation of the plane with the given characteristics. y – 4 The plane passes through the point…

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Q: Find the coordinates of a point that is located five units in front of the yz-plane, three units to…

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Q: trahedron enclosed by the coordinate planes and the plane x+2y+z= 8.

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Q: Give a geometric description of the set of points in space whose coordinates satisfy the given pair…

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Q: 17. projection of the line Find the equation of the 3x-y + 2z-1=0 = x +2y-z-2 on the plane 3x +2y +…

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Q: Tangent of x²2/3 + y2/3 + z²/3 = a²/3 surface at any point (Xo , Yo ,Zo ) Show that the sum of the…

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Q: det äLt)=cost ēit test(2-t-cost) ēga Show that the plane defermine ď is a plane curve, and that…

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Q: Find the maximum and minimum values of ƒ(x,y,z)=x^2+y^2+z^2 on the curve on which the cone z^2 =…

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Q: The prism is bisected by the xz-plane. The point P = (5, 4, 5). Determine the equation of the plane…

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Q: Find the coordinates of the point. The point is located nine units in front of the yz-plane, eight…

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Q: The Line LI : %3D Plane X+ 3y-Z= -3 , P and intersect the %3D Lind the point of intersection Irne in…

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Q: II. Find an equation of the plane containing the points (0,2,-1) and (1,-1,-1) and perpendicular to…

A: we have to find the equation of plane.  Let, a,b c are the direction ratios of required plane 1.

Q: Find the coordinates of the point. The point is located in the yz-plane, seven units to the right…

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Q: Three intersecting planes Describe the set of all points (if any) at which all three planes x + 3z =…

A: Consider the planes, The objective is to describe the set of all points at which all above three…

Q: Relationship between r and r' Consider the parabola r(t) = ⟨at2 + 1, t⟩, for - ∞ < t < ∞,…

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Q: The plane perpendicular to the x-axis at (–2,0, 0) : The plane perpendicular to the y-axis at (0,…

A: the equation is  (x1,y1,z1)=passing points a(x-x1)+b(y-y1)+c(z-z1)

Q: 1) Let A = (0, 0, 0) and B = (2, 0, 0). a) Find a point C in the xy-plane that is a distance 2 from…

A: a) C is a point on the xy-plane with a distance of 2 from both A and B.Consider C = (x, y,…

Q: Find a parametrization of the line in which the planes x-2y 2z = 2 and -3x + 3y-3z 0 intersect. Find…

A: The given planes are  x-2y+2z=2                   1-3x+3y-3z=0             2 From equation (2), we…

Q: Cone: x2 + y2 - z2 = 0, Plane: x + 2z = 4

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Q: A student is graphing the function y= Vz+4 on the coordinatė plane. Which statement will help the…

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Q: quation of the tangent plane to the surface at the given point. y + z2 = 9, (1, 2, 2) of symmetric…

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Domain

In simple mathematics terms, a domain is defined as the space on which all the possible inputs for the function will lie. It can also be expressed as a set of departures for the required function.