Good day guys, i'm starting with linear algebra but i got this problem that to be honest i don't know where to start:
"Let $A,B,C$ be not null vectors such that the angle formed by $A$ and $C$ equals to the angle formed by $B$ and $C$, then the vector $C$ is orthogonal to the vector $||B||A - ||A||B$."
I would appreciate so much if you guys could help me with this proof. Thank you.
In terms of the scalar product, the angle between $A$ and $C$ and between $B$ and $C$ being equal means that $$\langle A, C \rangle / ||A|| = \langle B, C\rangle / ||B||.$$ Using this, compute $$\langle ||B||A - ||A||B , C \rangle = ||B||\langle A, C \rangle - ||A||\langle B, C\rangle = 0.$$