Orthogonality question

Question

Been stuck on this one:

If $\vec{x}$ is orthogonal to $\vec{u}$ and $\vec{v}$ then $\vec{x}$ is orthogonal to $\vec{u}-\vec{v}$.

Any hints?

Answer 1

Since $\vec{x}$ is orthogonal to both $\vec{u}$ and $\vec{v}$, the equations $$ \langle \vec{x},\vec{u}\rangle=0\qquad\langle \vec{x},\vec{v}\rangle=0\tag{1} $$ hold. Furthermore, the general properties of inner-products give us the equation $$ \langle\vec{x},\lambda_1\cdot\vec{u}+\lambda_2\cdot\vec{v}\rangle =\lambda_1\cdot\langle\vec{x},\vec{u}\rangle+\lambda_2\cdot\langle\vec{x},\vec{v}\rangle\tag{2} $$ whenever $\lambda_1,\lambda_2\in\Bbb R$. Can you use equations (1) and (2) to solve your problem?