(projections by calculus) The dot product of … with itself is

(projections by calculus) The dot product of $\mathbf{B}-t\mathbf{A}$ with itself is $|\mathbf{B}{|}^2-2t\mathbf{A}\cdot \mathbf{B}+t^2|\mathbf{A}{|}^2$. (a) This has a minimum at $t=$ ______. (b) Then $t\mathbf{A}$ is the projection of ______. A figure showing $\mathbf{B},t\mathbf{A}$, and $\mathbf{B}-t\mathbf{A}$ is worth 1000 words.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §11.2 Planes and Projections, #43