… is perpendicular to a plane and … is along a line

$\mathbf{N}$ is perpendicular to a plane and $\mathbf{V}$ is along a line. Draw the angle $\theta$ between the plane and the line, and explain why $\mathbf{V}\cdot \mathbf{N}/|\mathbf{V}||\mathbf{N}|$ is $\sin \theta$ not $\cos \theta$. Find the angle between the $xy$ plane and $\mathbf{V}=\mathbf{i}+\mathbf{j}+\sqrt{2}\mathbf{k}$.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §11.2 Planes and Projections, #18