Determine the distance from point P(–2, –1, 1) to the plane [x, y, z] = [4, –1, 6] + t[1, 6, 3] + s[–2, 3, 1].

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 a function r(t) that describes the line through P(3,4,5) that is orthogonal to the plane 2x-z=4

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Q: Find the projection of d on the plane P if a 4 and P has the equation x + y+ z = 1.

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Q: Find the mapping that reflects the plane in the line x+y-1=C O (x,y)>(y+1,x+1) (x,y)→(-x+1,-y+1) O…

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Q: Let L be a line which passes through the point (-7,0, 1) and is perpendicular to the plane x + 2y +…

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Q: Find the point on the plane x – 2y + 3z = 6 that is closest to the point (0,1,1).

A: First we find out normal vector to the given plane than find out the point that is closed to the…

Q: Find the point on the plane x – 2y + 3z = 6 that is closest to the point (0, 1, 1).

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Q: Find the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1, 2), C =…

A: Directly use the formula for the equation of plane when 3 points are given.

Q: F = (x² – y², y² – 2², z² – x²), S is the part of the plane r + y + z = 1 in the first octant.

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Q: (A) Determine the plane that contain the point A(2,5,1) and the line L, t=x-2, 3t =y-1, t=z+1

A: The equation of the line L given in the parametric form, that is, Thus, the above equation of the…

Q: Determine the equation of the plane that contains the points P(1, –1,0), Q(1, 1, –1) and R(-1,0, 1).…

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Q: find the trace of the plane in the given coordinate plane.  −x + y = 4, xz

A: Find trace of the plane in the coordinate plane -x+y=4 in xz

Q: Find an equation of the plane. The plane that passes through the point (2, 3, 5) and contains the…

A: We need to find the equation of the plane. x=5t , y=2+t , z=3-t Plane passes through (2,3,5) the…

Q: Determine the image of the upper half plane {z € C : Im(z) > 0 }

A: In this question, we use the transformation under g(z) we find the image set.

Q: Let T be the plane x – y – z = 7. Which of the following planes are orthogonal to T and pass through…

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Q: ind the equation of the plane passing through the three points A = (-1,0, 2), B = (2, –1,2), C = (3,…

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Q: Find the point on the plane x+y+z=-46 that is closest to the point (1,1,1).

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Q: Let L be a line which passes through the point (-3,0, 1) and is perpendicular to the plane r + 2y +…

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Q: Show that the curve a(t) = (t,4,²) lies in a plane and find the equation of this plane.

A: Solution:- αt=t,1+tt,1-t2t t,1t+1,1t-t=x,y,z It is clear that, t-1t-1+1t-t=x-y+z x-y+z=-1

Q: If possible, determine the vector, parametric,symmetrical and reduced line r that: a) passes…

A: Given the lines S1:t·1,2,3, t∈ℝ and S2:y=x+2, z=x. To find a line L passing through the point P1,1,1…

Q: Which of the following is the equation of the plane that is tangent to the surface 2 x2 + y? x z –…

A: Given: 2x2+y2-xz-3z=1

Q: Find a plane containing the point (-3,3,-3) and the line F(t) = +t

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Q: Find the point on the plane x+ 2y - z 12 that is nearest the origin.

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Q: find the trace of the plane in the given coordinate plane.  3x + 4z = −2, xy

A: To find the trace of the plane in the given coordinate plane  3x+4z=-2 --- (1) in the xy plane

Q: Find an equation of the plane through (– 1, 3, –4) that is perpendicular to the planes 8x – y + z =…

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Q: a) Evaluate f, (2x – z) dS, where S is part of the plane 3x + 2y + z = 1 that lies above the…

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Q: Find all planes in R3 whose intersection with the xy-plane is the liner( t) = t ( 2, 1, 0) .

A: GIVEN  Intersection with the xy-plane is the line r( t) = t ( 2, 1, 0)

Q: Show there is no plane perpendicular to n=i+jn=i+j that passes through points P(1,2,3)P(1,2,3) and…

A: This is the rough image Point P(1,2,3) and Q(2,3,4) are the points on the plane…

Q: The distance from the point P(1,2,3) to the plane x +y + z = 0 is

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Q: Find the symmetric equations of the L that is parallel to the two planes x+y-z= 2 and z + 3y+ z =…

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Q: is ndicular tò the plane containing the points: A = (-1,1, 2) B = (-3, 2, 1) C = (5 – 1, 1) the

A: Given that the plane is passes through the point A, B and C. We have to find the normal vector of…

Q: The equation of the plane that passes through the point P(1,0,0) and has the vector n = (0,0, 1) as…

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Q: Let P be the plane with equation x-2y-2z = -11. Find four distinct points on P at distance 6√2 to…

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Q: 12. a. Use vectors to show that the distance from P;(x1, Y1, Z1) to the plane Ax + By + Cz = D is…

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Q: Evaluate x2 zdS, where S is the part of the S plane z = 1+ 2x + 3y above the rectangle [0, 2] × [0,…

A: We have to evaluate ∬Sx2zdS, where S is the part of the plane z=1+2x+3y above the rectangle 0,2×0,3.…

Q: Which of the following are equations for the plane through the points P(1, 1, -1), Q(3, 0, 2), and…

A: First find PQ and PR , two lines on the plane

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: Let R be the triangle in the xy-plane with vertices (0,0),(0, 1),(1,0). Find C[n(2 – x – y) dA.

A: Given vertices of triangle now we have to find the area as shown below

Q: Find an equation for the plane passing through the origin that is parallel to the tan- gent plane to…

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Q: Find the projection of a on the plane P if a 4 and P has the equation x + y + z = 1.

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Q: Find a parametrization r(t,s) for the plane which contains the point P(-9, 7, 1) and the line…

A: Use plane concept

Q: Consider the points P(1,0, 3) and Q(5, 2, −1) and the line 7 : y = t 2 = 9+t 1. Find the the…

A: This is a question of vector algebra and geometry.

Q: Let L be a line which passes through the point (-5, 0, 1) and is perpendicular to the plane r + 2y+…

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Q: Find the equation of the plane passing through the three points A = (-1,0,2), B= (2, –1,2), C = (3,…

A: Given problem:- Find the equation of the plane passing through the three points A = (- 1, 0, 2) B =…

Q: a) Show that the line I = 0, y = t, z =t where te R a) lies in the plane 6z+4y–4x = 0 b) is parallel…

A: In the question it is asked to solve for the planes having equation given below.

Q: Let P be the plane with equation x-2y+2z =-10. Find four distinct points on P at distance 6 v2 to…

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Q: Find the distance of point K(1,0,1) to the plane generated by the points P(2,3,1), Q(3,-1,4) and…

A: All options are wrong i am giving Right answer with complete formula and procedure for easy…

Q: Find the equation of the plane passing through the three points A = (-1,0,2), B = (2,–1,2), C = (3,…

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Q: Let P be the plane 2x − 3y + 5z = 0. Find a surjective map T : R^3 → P such that T(1,2,3) =…

A: Given below the detailed solution