Answer to Question #143591 in Linear Algebra for Alex

Question

Orthogonal
Is it true that if x is orthogonal to v and w, then x is also orthogonal to v − w. Why so?

Answer 1

It is well known that the scalar product is a distributive with regard to the sum and the difference of vectors:

$a\cdot(b\pm c)=a\cdot b\pm a\cdot c$.

The vectors $a$ and $b$ are ortogonal if and only if the scalar product $a\cdot b$ is equal to 0.

If  $x$ is orthogonal to $v$ and $w$, then $x\cdot v=0$ and $x\cdot w=0$. Therefore,

$x\cdot(v-w)=x\cdot v- x\cdot w =0-0=0,$

and we conclude that $x$ is also orthogonal to $v − w.$