Projection of a Vector

The idea

The projection of a vector $\overrightarrow{AB}$ onto a directed line $l$ is how much of $\overrightarrow{AB}$ points along $l$'s direction, its "shadow" cast onto that line, scaled by $\cos\theta$ where $\theta$ is the angle between them.

Worked example: a special angle

What is the projection of $\overrightarrow{AB}$ on a directed line $l$ if the angle between them is $\theta = \pi$ (180°)?

Solution: An angle of $\pi$ means $\overrightarrow{AB}$ points exactly opposite to the direction of $l$. The projection is then a vector of the same length as $\overrightarrow{AB}$ but pointing the other way along $l$, exactly $\overrightarrow{BA}$, the negative of the original vector.