Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 and the line L in R³ spanned [9] by the vector 3 Let S R3 R³ denote the reflection through the plane P: it takes a vector in R³ and : transforms it into its mirror image, the mirror being the plane P. Let T: R³ → R³ denote the 90° rotation around L of your choice (i.e. you can choose if the rotation is clockwise or counterclockwise). Both S and T are linear transformations (you don't have to prove that). Find the matrix A such that (ToS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21] are perpendicular to the line. They are also perpendicular to each other. The vectors and -7 30 M2x2 T Maxz DB [-] B DB "K [T] B-B B-world

Let P(3,-1, 4) and Q(10, 13,9) = (2, 5, -1) and v = (7,-1,2) l be the line through P along direction u and II be plane through Q perpendicular to v. u = Find parametric equations for l. Find an equation, in the usual coordinates x, y, z, for II. Find the coordinates of the intersection of and II.

1. Which value of k will make the lines: r = (1,2) +s(2,3), s e R and 9x + ky = 0 parallel? 2. Determine the parametric cquations of a line passing through the point P(3, 2, –1) and with a direction vector perpendicular to the line r - (2,-3,4) + s(1, 1,–2), s e R. 3. Determine the Cartesian equation of the plane having x-, y-, and z-intercepts of -3, 1, and 8 respectively.

Let ü = (-2, –3) and d= (1,4). (a) What is the parametric equation of the line through ũ with direction d? (b) What are the symmetric equations of the line through ũ with direction d? (c) What is the y = mx + b form of the line through ï with direction d?

If a line is parallel to the vector 6i +7j+ 2k and passes through the point P = (5, 1,7), then the parametric equation of the line is: O a. X=5+ 62, Y= 1+ 7A, Z= 7+2\ + O b. X=-5+ 6A, Y= -1+ 7), Z= -7+2\ O c. X=-5-62, Y=-1-7), Z= -7-2A O d. X=5-6A, Y= 1-7A, Z= 7-2) MacBook Air esc & 1 2 € 4 5 6. 8. Q E T Y A D G H J > C V %3D ol option command command option