I am working on a vector question for my physics course and am having some trouble with part C.
The question reads,
"Let $\vec{a}$ = $\langle -1, 3, 0 \rangle$ and $\vec{b}$ = $\langle -1, 3, 6 \rangle$.
a) Find the scalar projection $l$ of $\vec{b}$ onto $\vec{a}$.
b) Find the vector $\vec{c}$ in the direction of $\vec{a}$ with length $l$.
c) Show that $(\vec{b}-\vec{c})\cdot\vec{a} = 0$. Explain why this must be the case for any vectors $\vec{a}$ and $\vec{b}$."
For parts a) and b), I got answers of $\sqrt{10}$ and $\langle -1, 3, 0 \rangle$ respectively. I was also able to prove that the equation was true for the given values of $\vec{a}$ and $\vec{b}$, but I have no clue as to where to start in proving that $(\vec{b}-\vec{c})\cdot\vec{a} = 0$ for all vectors $\vec{a}$ and $\vec{b}$.
Any help would be greatly appreciated. Thanks!