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What are Orthogonal Projections?
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If $\mathbf{u}$ and $\mathbf{a}$ are vectors in $R^{n}$, and if $\mathbf{a} \neq 0$, then $\mathbf{u}$ can be expressed in exactly one way in the form $\mathbf{u}=\mathbf{w}_{1}+\mathbf{w}_{2}$, where $\mathbf{w}_{1}$ is a scalar multiple of $\mathbf{a}$ and $\mathbf{w}_{2}$ is orthogonal to $\mathbf{a}$ .
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