Problem 23. Consider the plane P in R³ defined by the equation x+2y+3z = 0 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. 3 -21 are perpendicular to the line. They are also perpendicular to each other. • The vectors and -7 0 30

can you make problem more clearer to understand Find vectors u and v such that the parametric equation of the line y= mx + b can be expressed as [yx] = tu + v.

Which of the following is a direction vector for the plane with Cartesian equation 3x -2y - 5z + 3 = 0?

. 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 .

Find the vector and parametric equations of the line through the points A = (2,4,7) and B = (−1,5,3).

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.

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