Vector projection

The shadow one vector casts onto another, as a full vector pointing along the second one.

The vector projection $\text{proj}_{\mathbf{u}}\mathbf{v} = \dfrac{\mathbf{v}\cdot\mathbf{u}}{|\mathbf{u}|^2}\,\mathbf{u}$ is a genuinely new vector, shorter than $\mathbf{v}$ itself, pointing exactly along $\mathbf{u}$ — the fraction out front is a scalar that stretches or shrinks $\mathbf{u}$ down to the shadow's length. It is the shadow $\mathbf{v}$ would cast on $\mathbf{u}$ 's line if a light shone straight down onto $\mathbf{v}$ from above that line.

Examples

Projecting a force vector onto a displacement direction isolates the part of the force that actually contributes to work.