20 Let A 20 6 B 31 4 and C 1 2 3 a Find the projection o

2.0) Let A = (-2,0, 6) , B = (3,-1, 4) and C = (1, 2, 3).

(a) Find the projection of the vector from A to B onto the direction of the vector from A to C.

(b) Find a unit vector that is perpendicular to the plane containing the points A, B and C.

(c) Find parametric equations of the line passing through points A and C.

(d) Find an equation of the plane that passes through C with normal vector given by the
vector from A to B.

(e) Find the area of the parallelogram spanned by the vectors
AB and AC.

Solution

a) AB=(3+2,-1-0,4-6)=(5,-1,-2) AC=(-3,2,-3) proj =(AB.AC)/|AC| =(-15-2-6)/sqrt(22) =-23/sqrt (22) c) x = 3t+1 y = 2t+2 z= -2 t + 3 d) x = 5 t+1 y = -1 t+2 z= -2 t + 3 e)/ AB * AC / = 14.86