Let $a, b$, and $c$ be unit vectors such that $a\cdot b=1/3 , b\cdot c=1/7$ and $a\cdot c=1/10$. Find $(a+b+c)\cdot (a−b)$

Question

So from this, I managed to find out the magnitude of $a,b,$ and $c$ respectively, finding out my previous parts:

$3a\cdot 7b = 3$

$a\cdot (b−c) = 7/30$

But I can't seem to find $(a+b+c)\cdot(a−b)$ :/ Any clues?

Answer 1

Clues:

Use properties of dot product such as $(d+e)\cdot f=d\cdot f+ e\cdot f,$

$ g\cdot h=h\cdot g$, and if $i$ is a unit vector then $i\cdot i=1$.