Lin. Alg. Projections conceptual question

[Jarvis323]

Homework Statement

16. Suppose P is the projection matrix onto the line through a.
(a) Why is the inner product of x with Py equal to the inner product of Px with y?
(b) Are the two angles the same? Find their cosines if a = (1;1;¡1), x = (2;0;1),
y = (2;1;2).
(c) Why is the inner product of Px with Py again the same? What is the angle
between those two?

Homework Equations

Px = (a^t x / a^t a)a
Py = (a^t y / a^t a)a

The Attempt at a Solution

I tried to multiply out the general vectors of size n and then do their dot product, but it got way to complicated.

I'm not sure which angles they want me to find the angles between. Obviously the angle between Px and Py is 0. But the angles between x and y could be arbitrary ( not talking about the specific ones in part b ), and I can't see which angles between which vectors should make it obvious that part a is true.

 

[LCKurtz]

Homework Statement

16. Suppose P is the projection matrix onto the line through a.
(a) Why is the inner product of x with Py equal to the inner product of Px with y?

Is P given to be an orthogonal projection? If so try writing $x = u_1 + v_1$ where $u_1$ is on the line and $v_1$ is perpendicular to it. Similarly for $y$. Then try working out those inner products, without getting down to the component level.