Problem 23. Consider the plane P in R³ defined by the equation x+2y+32= = O and the line L in R³ spanned 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 (TS)(x) = Ax for all vectors R³. Here is some information that you might find useful: The vector 2 is perpendicular to the plane. -21 The vectors 3 and 0 -7 30 are perpendicular to the line. They are also perpendicular to each other. Finally, here is a sugestion: don't hesitate to use a 3D graphing calculator (like Desmos) to help you picture the plane and the line.

. Let A= (2, 3, 1) and B= (4, -1, 2) and C = (6, 2, 3). Find the equation of the line in parametric form passing through C that intersects AB is perpendicular to AB .

The parametric vector v=(6-t)i+(9t+8)j represents a straight line. Find the equation of this line

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.