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I am given the vectors $\mathbf{w}, \mathbf{v}, \mathbf{u}$ in $\mathbb{R}^n$ such that $\mathbf{u} \neq 0$ and $\mathbf{w} = \mathbf{v} - \frac{\mathbf{u}\,\bullet\,\mathbf{v}}{\|\mathbf{u}\|^2}\bullet\mathbf{u}$. I am asked to show that the vector $\mathbf{w}$ is orthogonal to $\mathbf{u}$.

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So far, I have written out the definition of orthogonal: two vectors are orthogonal if and only if their dot product is zero.

So what we need to prove is $\mathbf{w}\bullet\mathbf{u} = 0$ where $\mathbf{w}\bullet\mathbf{u}$ is defined as $\mathbf{w}^T\bullet\mathbf{u}$.

However, I have been stuck on this problem for about an hour and haven't made any significant progress from here. How do we go about proving that the vectors are orthogonal?

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1 Answer

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We have: $$\mathbf w\bullet\mathbf u=\left(\mathbf{v}-\frac{\mathbf{u} \bullet \mathbf{v}}{\|\mathbf{u}\|^2}\bullet \mathbf u\right)\bullet \mathbf u$$ Distribute the $\mathbf{u}$: $$\mathbf{v} \bullet \mathbf{u}-\frac{\mathbf{u} \bullet \mathbf{v}}{\|\mathbf{u}\|^2}\bullet \mathbf u\bullet \mathbf u$$ We know $\|\mathbf u\|^2=\mathbf u\bullet\mathbf u$, so cancel: $$\mathbf v \bullet \mathbf u-\mathbf u\bullet\mathbf v=0$$

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