$\widehat{W}^{T}\varphi w_{1} = 0$?

Question

Let $W \in \mathbb{R}^{m\times m}$ be orthogonal matrix with first column $w_{1}$.

Let $\widehat{W}$ be the other columns in $W$.

So $W$ can be written as $\begin{bmatrix} w_{1} & \widehat{W} \end{bmatrix}$

Can someone help me to explain why $\widehat{W}^{T}\varphi w_{1} = 0$, due to orthogonality of $W$ where $\varphi$ is a scalar?

I tried $\widehat{W}^{T}\varphi w_{1} = \varphi (\widehat{W},w_{1})$, but don't know what to do next.

Answer 1

By the definition of orthogonal matrix, $$I=W^TW = \pmatrix{w_1^T \\ \widehat W^T}\pmatrix{w_1 & \widehat W}=\pmatrix{w_1^Tw_1 & \dots \\ \widehat W^Tw_1 & \dots},$$ so $\widehat W^Tw_1=0$.