Exercise 14

Show that the definition of orthogonal projection into a line does not depend on the spanning vector: if $\vec{s}$ is a nonzero multiple of $\vec{q}$ then $(\vec{v}\mathbin{\smash{\bullet}}\vec{s}/\vec{s}\mathbin{\smash{\bullet}}\vec{s}\,)\cdot\vec{s}$ equals $(\vec{v}\mathbin{\smash{\bullet}}\vec{q}/\vec{q}\mathbin{\smash{\bullet}}\vec{q}\,)\cdot\vec{q}$.