So the question I have is for:
Answer the following with either an explanation, a diagram or a proof:
If $a\cdot b=a \cdot c$ , what is the relationship between $b$ and $c$? If $a\times b =a\times c$ , what is the relationship between $b$ and $c$?
So $a$ is orthogonal to $(b-c)$ as if $a\cdot b=a \cdot c$, then $a\cdot b=a \cdot c= 0$, then $a\cdot(b-c) = 0$, therefore $(b-c)$ is orthogonal to $a$, but how do I express the relationship between $b$ and $c$?
Here's what else I got:
$a\cdot b = |a||b|\cos(ab)$
$a\cdot c = |a||c|cos(ac)$
$|a||b|\cos(ab) - |a||c|\cos(ac) = 0$
$|a|(|b| \cos(ab) - |c|\cos(ac)) = 0$
$|b| = |c|\cos(ac) / \cos(ab)$
I'm not sure what I'm missing for the dot product question. For the cross product question, which properties would be used to solve it?