A new vector
Define the orthogonal projection of $\mathbf{b}$ onto $\mathbf{a}$ to be $\text{orth}_{\mathbf{a}} \mathbf{b}= \mathbf{b} - \text{proj}_{\mathbf{a}} \mathbf{b}$.
Show that the vector $\text{orth}_{\mathbf{a}} \mathbf{b}$ is orthogonal to $\mathbf{a}$.
More Projections
Suppose $\mathbf{a}$ and $\mathbf{b}$ are non-zero vectors.
When does $\text{proj}_{\mathbf{a}} \mathbf{b} = \text{proj}_{\mathbf{b}} \mathbf{a}$?
Orthogonality
Show that if $\mathbf{u} + \mathbf{v}$ and $\mathbf{u} -\mathbf{v}$ are orthogonal, then $\mathbf{u}$ and $\mathbf{v}$ must have the same length.