Hi guys Im stuck on a question.
Given $\{u,v,w\}$ orthonormal set, prove that $\{u+2v+w,u-v+w,u-w\}$ is an orthogonal set. I know that im supposed to prove
$$\langle u+2v+w,u-v+w\rangle=0$$ $$\langle u+2v+w,u-w\rangle = 0$$ $$\langle u-v+w,u-w\rangle= 0$$
but I have no idea on how to do it. Thanks in advance.
Recall that $$\langle x+y,z\rangle=\langle x,z\rangle+\langle y,x\rangle$$ that is $$(x+y)\cdot z=x\cdot z+y\cdot z$$
You know that $$\langle u,v\rangle =\langle u,w\rangle =\langle w,v\rangle =0$$
$$\langle v,v\rangle =\langle u,u\rangle = \langle w,w\rangle =1$$
EXAMPLE
$$\begin{align} \langle u - v + w,u - w\rangle &= \langle u - v,u - w\rangle + \langle w,u - w\rangle \\ &= \langle u,u - w\rangle + \langle - v,u - w\rangle + \langle w,u\rangle + \langle w, - w\rangle \\ &= \langle u,u\rangle + \langle u, - w\rangle + \langle - v, - w\rangle + \langle - v,u\rangle + \langle w,u\rangle + \langle w, - w\rangle \\ &= \langle u,u\rangle - \langle u,w\rangle + \langle v,w\rangle - \langle v,u\rangle + \langle w,u\rangle - \langle w,w\rangle \\ &= 1 - 0 + 0 - 0 + 0 - 1 = 0 \end{align}$$
One can write things more concisely, of course $$\left( {u - v + w} \right)\cdot \left( {u - w} \right) = {u\cdot u} - u\cdot v + w\cdot u - w\cdot u + v\cdot w - {w \cdot w} = 1 - 1 = 0$$
You can do the same for the others. $$(u + 2v + w)\cdot(u - w) = {u^2} + 2v\cdot u + w\cdot u - w\cdot u - 2v\cdot w - {w\cdot w} = 1 - 1 = 0$$
$$\begin{align}(u + 2v + w)\cdot(u - v + w) &=u\cdot u + 2v\cdot u + w\cdot u - u\cdot v \\& - \;\;2{v\cdot v} - v\cdot w + w\cdot u \\&+\;\;2v\cdot w + {w\cdot w} \\&= 1 - 2 + 1 = 0\end{align}$$