Vector Projection

The Vector Projection of a Vector $\mathbf{a}$ onto another vector $\mathbf{b}$ is the component of $\mathbf{a}$ that points in the direction of $\mathbf{b}$. You can think of it as the "shadow" $\mathbf{a}$ casts on the line through $\mathbf{b}$.

It's calculated using the Dot Product:

$\text{proj}_{\mathbf{b}}\mathbf{a} = \frac{\mathbf{a} \cdot \mathbf{b}}{\mathbf{b} \cdot \mathbf{b}} \mathbf{b}$

The scalar projection, $\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{b}|}$, gives just the (signed) length of that shadow, where $|\mathbf{b}|$ is the Vector Magnitude of $\mathbf{b}$.