Consider the plane given by the scalar equation -2x + 4y – 3z = 3 and the line given by the parametric representation L1: (3, -2,5) + t(-3,6, -3). Find their point of intersection. Answer: FORMATTING: Enter your answer in the form of (x, y, 2). The components must be exact, namely, fractions, not decimal approximations.

Q: Find the parametric equations for the line through the point of intersection of the lines L,:x = 2…

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Q: Find the equation of the plane in the form z=c+mx+ny containing the line in the xy-plane where x=3,…

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Q: Find the real number a such that the line of x = 1, y = 2-1, z=3+t, the plane of equation parametric…

A: Given parametric equations are x=t , y=2-t, z=3+t Where t∈ℝ and  Plane is αx+5y+z-10=0

Q: Let C be the curve of intersection of the hyperboloid x2−y2+z2 = 4 and the plane 2y−z = 0. Find…

A: Let C be the curve of intersection of the hyperboloid x2 - y2 + z2 = 4 and the plane , 2y - z = 0.…

Q: Find the scalar equation for the plane passing through the point P=(-5, -2, –4) and containing the…

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Q: Find an equation of the plane that contains the intersection points of the ellipsoid 9x? + 4y² + 9z²…

A: GIVEN : Ellipsoid 9x2 +4y2 + 9z2 = 36

Q: -Determine the equation of the trace of this plane in the xz plane given 2x + y − z = 4.…

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Q: Consider the planes II : 3z – y+ 2z=1, Il2 : -9z + 3y – 6z = 24 and the line of parametric equations…

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Q: Find the scalar equation for the plane passing through the point P-(5, 3,-5) and containing the line…

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Q: What is the parametric equation for the line of intersection of the planes x + y - z = 2 and 3x - 4y…

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Q: Find the scalar equation for the plane passing through the point P=(4, -5, 3) and containing the…

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Q: Find parametrizations for the line in which the planes 3r - 6y-2z = 3, and 2x+y-2z = 2 intersect.

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Q: Give a scalar equation for a plane at distance 9 from the plane P with equation -2x-2y+z = 4.

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Q: Find a vector parametric equation for the ellipse that lies on the plane 5y−3x+z=9 and is inside the…

A: Given that: 5y-3x+z=9 Let x=u cos v, y=u sin v So z=9+3x-5y  =9+3u cos v-5u sin v

Q: (a) Prove that the lines with parametric equations: x = 1+t, y = 1+ 6t, z = 2t and x = 1+ 2s, y = 5…

A: aLet the L1 be the line:x=1+t, y=1+6t,z=2tand L2 be the line:x=1+2s, y=5+15s, z=-2+6sThe direction…

Q: . Consider the plane P : x +y + z = 2 and point A(0,1, 2). A line L has the parametric equations L:…

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Q: x = −1+5t y = 3-s-2t z= −2+s

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Q: Find a parametrization of the line in which the planes x+y +z= -2 and y +z= 6 intersect.

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Q: Where does the parametric line ( 2 + 2 t , t , t ) intersect the plane x+y-3z=4?

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Q: 2. Find a parameterization of the line formed by the intersection of the following planes: P1:2x – y…

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Q: Find the scalar equation for the plane passing through the point P=(−4, 2, 10) and containing the…

A: Explained below

Q: Find an equation of the tangent plane to the following parametric surface, r(u, v) = (u2 + 6) i+ (v³…

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Q: Find the scalar equation of the plane that contains the line x+1 -3=. and is parallel to x+2=+4_:+2…

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Q: Determine the intersection of the hyperbolic paraboloid z = (y2/b2) − (x2/a2) with the plane bx + ay…

A: Given The hyperbolic paraboloid equation is z=y2b2-x2a2 and the plane bx + ay − z = 0 .

Q: Consider the plane P with the Cartesian equation 7x+5y + 5z = 33. a) Express the above Cartesian…

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Q: Complete the parametric equations for the line where the planes – 8z – 2y – 5z = 0 and - 4x + 2y –…

A: Given: The planes are -8x-2y-5z=0 and -4x+2y-9z=0 To find the parametric equation xt, yt, zt

Q: Find a parametric representation of the line through the point P(3,1,-4) and perpendicular to the…

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Q: Complete the parametric equations for the line where the planes 8x + 11y + 4z 13x + 2y + 3z = 5…

A: This question can be solved using the concept of lines and planes in three dimensional geometry.

Q: Given the curve of intersection of the x² + 4y² -z = 0 which is a paraboloid and the plane 2x - z =…

A: Solution:-

Q: For planes 2x+ y – z = 2 and 2x + 3y + 5z = 4. %3D Write the parametric equations of the line at…

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Q: Find the equation for the sphere centered M(2,-3,2) tangent to the D:6x-3y+2z-11=0 plane

A: we have to find the equation for the sphere centered 2,-3,2 and tangent to the plane 6x-3y+2z-11=0.

Q: The surface z? = x² + y? – 4x + 5 remains unchanged when rotated about the line L in R Which of the…

A: The given surface is z2=x2+y2-4x+5, which is as shown below: This is a hyperboloid paraboloid.…

Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? A) (2 +…

A: The direction ratio of the normal vector to the plane is (3, 2,1 ) So the perpendicular line to the…

Q: Find the point on the curve of intersection of the paraboloid x² • y² + 2z=4 and the plane x-y+ 2z…

A: This question can be solved by options because we have only two equations and three variables.

Q: Consider the following equation of a plane in R³: -3z – 2y – 3z = -2 Convert your plane to vector…

A: Find the vector parametric form of the plane: -3x - 2y - 3z = -2

Q: ). Find a parametric representation for the line of intersection of the following two planes: I - y…

A: We use cross product to find the direct vector for the line.

Q: Find the scalar equation for the plane passing through the point P-(0,-3, 2) and containing the line…

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Q: 1. a) Given the lines L₁ : 7 = (1,2,−4) + t(k+ 1,3k + 1, k − 3) and L₂: x=2-3s, y = 1-10s, z = 3-5s,…

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Q: Find an equation for the plane that passes through the point (1, 2, 3) parallel to u = 2i + 3j + k…

A: Given:u = 2i + 3j + k and v = i - j + 2kFirst find cross product

Q: Find parametrizations for the line in which the planes 3r- 6y – 2z = 3, and 2x+y-2z = 2 intersect.

A: Given Equation of planes 3x-6y-2z=3 and 2x+y-2z=2

Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? a)…

A: The direction ratio of the normal vector to the plane is (3, 2, 1) So the direction ratio of the…

Q: Find parametric equations for the line through (- 3,1,2) perpendicular to the plane 7x + 9y + 9z =…

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Q: 9) Find a parameterization of the line formed by the intersection of the following planes: P,: 2x -…

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Q: Complete the parametric equations for the line wherekche planes 2=0and-2z + 2y + 2z -4 intersect.…

A: The equation of the planes are: P1: 2z = 0 ⇒ z = 0                                  .........1 P2:…

Q: a) Prove that the lines with parametric equations: x = 1+ t, y = 1 + 6t, z = 2t and x = 1+ 2s, y = 5…

A: Given that: Line L1:  x=1+t, y=1+6t, z=2tLine L2:  x=1+2s, y=5+15s, z=-2+6s

Q: Give an equation of the plane whose trace in the yz-plane has equation y = 6z, and in the xy-plane…

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Q: Find the point at which the line with parametric equations x = 3 + 4t, y = 5 – 2t, z = 4 + 7t…

A: According to question,Given equation of line in parametric form,x = 3+4t,y = 5-2t,z= 4 +7t,Equation…

Q: Find the parametric equations for the line of the intersection L of the two planes. 6x – 3y+ z = 5…

A: Parametric equations of the line is :  x(t) = x0 + a t  y(t) = y0 + b t  z(t) = z0 + c t  Here ( x0…