Suppose … and

Suppose $\mathbf{I}=(\mathbf{i}+\mathbf{j})/\sqrt{2}$ and $\mathbf{J}=(\mathbf{i}-\mathbf{j})/\sqrt{2}$. Check $\mathbf{I}\cdot \mathbf{J}=0$
and write $\mathbf{A}=2\mathbf{i}+3\mathbf{j}$ as a combination $a\mathbf{I}+b\mathbf{J}$. (Best method: use $a$ and $b$ from Problem 27. Alternative: Find $\mathbf{i}$ and $\mathbf{j}$ from $\mathbf{I}$ and $\mathbf{J}$ and substitute into $\mathbf{A}$.)

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) · §11.1 Vectors and Dot Products, #28