Orthogonal Projection

Theorem

Let

• $\mathbf{u},\mathbf{a}$ : Vectors in $R^n$
• $\mathbf{a}\ne 0$ (is not the zero vector)

If

• ${\mathbf{w}}_1$ : Vector that is scalar multiple of $\mathbf{a}$
• ${\mathbf{w}}_2$ : Vector that is orthogonal to $\mathbf{a}$

Then

• $\mathbf{u}$ can be expressed in exactly one way in the form of $$ \mathbf{u}=\mathbf{w}{1}+\mathbf{w}{2}