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 $\mathbf{u}$.

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?


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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