Magnitude of the projection of one vector on another one, and viceversa

Question

When is the magnitude of the projection of vector $\mathbf{a}$ on vector $\mathbf{b}$ equal to the magnitude of the projection of vector $\mathbf{b}$ on $\mathbf{a}$?

Unsure when this is true.

Thanks.

Answer 1

For non-zero $a, b$: $$ \lvert a \cdot b/ \lVert b \rVert \rvert = \lVert a \rVert \lvert \cos \angle(a,b) \rvert \\ \lvert b \cdot a/ \lVert a \rVert \rvert = \lVert b \rVert \lvert \cos \angle(a,b) \rvert $$ Both are equal if the vectors are perpendicular or have same magnitude.

Answer 2

You need $\|\mathbf{a}\| = \|\mathbf{b}\|$ or orthogonality. See here