Projections

Let $\overrightarrow{PQ}=\bar a$ and $\overrightarrow{PR}=\bar b$ share initial point $P$, and let $M$ be the

foot of the perpendicular from $R$ to the line containing $PQ$ (so $\overrightarrow{PS}$, with $S=M$, is that

foot). Then:

• Scalar projection of $\bar b$ on $\bar a$ is $PS=|\bar b|\cos\theta=\dfrac{\bar a\cdot\bar b}{|\bar a|}$ -- described in the text as the "shadow" $\bar b$ casts on the line of $\bar a$ if the sun were directly overhead (i.e. shining perpendicular to $\bar a$). Likewise the scalar projection of $\bar a$ on $\bar b$ is $\dfrac{\bar a\cdot\bar b}{|\bar b|}$.
• Vector projection of $\bar b$ on $\bar a$ is that scalar length carried in the direction of $\hat a$:

Symmetrically, the vector projection of $\bar a$ on $\bar b$ is $\dfrac{\bar a\cdot\bar b}{|\bar b|^2}\bar b$.

These formulas are the standard tool for finding the foot of a perpendicular from a point onto a line (project

the vector to the point onto the line's direction vector and add back the starting point) and for splitting a …