Linear Transformation Question

Question

Suppose u = (2, 1, -1)/√6 and let P_u be the linear transformation that projects R³ onto the line through the origin in the direction of the unit vector u. Find all vectors v ∈ R³ with the property that P_u(v) = (4, 2, -2).

Answer 1

The formula for orthogonal projection onto a vector u is P_u(v)=[(u•v)/(u•u)]u.. Because u is a unit vector, the denominator is 1, and since (4, 2, -2) =(2√6)u, we can rewrite the problem as finding all v such that u•v=2√6. If we let v=(x,y,z), then (multiplying both sides of the resulting equation by √6), we get the equation

2x+y-z=12.

How you express the solutions to this equation is somewhat a matter of preference.