(a) For … and …, compute … and … and

(a) For $\mathbf{A}=\mathbf{i}+\mathbf{j}-4\mathbf{k}$ and $\mathbf{B}=-\mathbf{i}+\mathbf{j}$, compute $(\mathbf{A}\times \mathbf{B})\cdot \mathbf{i}$ and $(\mathbf{A}\times \mathbf{B})\cdot \mathbf{j}$ and $(\mathbf{A}\times \mathbf{B})\cdot \mathbf{k}$. By Problem 45 those are the areas of projections onto the $yz$ and $xz$ and $xy$ planes.
(b) Square and add those areas to find $|\mathbf{A}\times \mathbf{B}{|}^2$. This is the Pythagoras formula in space (Remark 2).

This exercise builds on another exercise in its source book, which isn't shown here.

Gilbert Strang, Calculus, 3rd edition (Wellesley-Cambridge Press; MIT OpenCourseWare RES.18-001) · §11.3 Cross Products and Determinants, #46