4. Prove that these lines are contained in the same plane. [x, y, z]= [1, 4, 0] + s[-2, 1, 3] [x, y, z]= [3, -4, -2] + t[0, -1, -2]

Q: 1. The plane containing the point (1,-5,8) and orthogonal to the line given b x=-3+15t, y=14-t,…

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Q: 4. For what value of h is vz in Span {v, v2} and for what value of h is {v1, V2, V3} linearly…

A: v1=1-32, v2=-39-6 and v3=5-7h

Q: Show that the line [x, y, z]= [5,-1,3]+[4,-5,2] lies on the planes 3x+2y-z = 10 and 7x+4y-4z=19.

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Q: A plane is perpendicular to [x, y, z] = [3, -30, 24] + s[2, 4, -2] and contains the point…

A: Step 1:​Plane is perpendicular to [x,y,z]=[3,−30,24]+s[2,4,−2]Direction of Line…

Q: Find the distance between the parallel lines in each pair. a) l₁: [x, y, z] = [2, 5, 4] + [2, 1, 3]…

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Q: Let R denote the rectangle with corners (−1, −2), (4, −2), (4,1), and (-1,1), and let P denote the…

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Q: Find the value of k so that the line [x, y, z] = [2, -2, 0] + t[3, 5, k] is parallel to the plane 4x…

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Q: What restrictions must be made on x, y, and z so that the triple (x, y, z) will represent points on…

A: The question asks to give restrictions must be made on x, y, and z so that the triple (x,y,z) will…

Q: 1. Do the following equations describe lines (L) or planes (P)? a) [x, y] = [1, 1] + s[2, 3] b) [x,…

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Q: 3. Find an equation for the plane in R³ containing the points P = (4, 1, 0) Q=(3, 2, 1) R = (5, 1,…

A: The given points are, P=(4,1,0)Q=(3,2,1)R=(5,1,1). Let an equation containing three points…

Q: Here let A, B, C, and D be four distinct points in the plane. Now supposedly that no three of them…

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Q: Determine if the line intersects the plane. If so, state the solution. [x, y, z][-1, -9,16]++[3, 3,…

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Q: 4. Find an equation of the plane that passes through the points P(1,3,2), Q(3,–1,6), and R(5,2,0).…

A: Given points P1,3,2, Q3,-1,6, R5,2,0

Q: 4. Find an equation for the plane in R³ spanned by the points (1,5, 1) and (1, 4, 2).

A: The two vectors are given as v1 = 151v2 = 142 Let xyz∈span{v1 , v2} Then xyz = mv1+nv2= m151+n142=…

Q: Let a = (3, −4, 9) (a) Find k such that b = (−2, k, −6) is parallel to a. (b) Find k such that b =…

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Q: . Show that the line connecting the planes x + 2y - 2z = 5 and 5x - 2y - z = 0 is parallel to he…

A: Explaination of the answer is as follows

Q: Which one of the points P(6, 9, -3), Q(8, -2, -8), R(-5, 8, 1) in 3-space is closest to the…

A: We have to find closest point to the xz plane from given points P, Q and R. Topic:- 3D vector.

Q: Determine the intercepts of each plane. a) [x, y, z] = [1, 8, 6] + s[1, –12, –12] + t[2, 4, -3]

A: As per policy, only the first question is answered as you have asked multiple questions asked in a…