How to determine $t$ for orthogonal to $u$ and $a$ in $\mathbb{R}^n$?

Question

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I understand that if $\mathbf{a}-\mathit{t}\mathbf{u}$ is orthogonal to $\mathbf{u}$ or $\mathbf{a}$, there must exist $(\mathbf{a}-\mathit{t}\mathbf{u})\bullet\mathbf{u}=0$ or $(\mathbf{a}-\mathit{t}\mathbf{u})\bullet\mathbf{a}=0$. However, I don't quite know in $\mathbb{R}^n$, how to determine $\mathbf{u}$ and $\mathbf{a}$. Is there any expressions of them that I can use to solve the problem?

Answer 1

Well, I'd consider $v = a - \langle u,a\rangle u$. Then by linearity in the 2nd argument, $$\langle u,v\rangle = \langle u,a\rangle - \langle u,a\rangle\langle u,u\rangle.$$ But $\langle u,u\rangle =\|u\|^2$ and by hypothesis $\|u\|=1$. Hence, $\langle u,v\rangle=0$.