Scalar projection

The signed length of one vector's shadow along another, $\text{comp}_{\mathbf{u}}\mathbf{v}=\dfrac{\mathbf{v}\cdot\mathbf{u}}{|\mathbf{u}|}$ .

The scalar projection of $\mathbf{v}$ onto $\mathbf{u}$ , $\text{comp}_{\mathbf{u}}\mathbf{v}=\dfrac{\mathbf{v}\cdot\mathbf{u}}{|\mathbf{u}|}$ , is the number that most work computations actually need — the length (with sign) of the shadow $\mathbf{v}$ casts onto $\mathbf{u}$ 's line. A negative scalar projection is meaningful, not an error: it means the shadow falls on the backward extension of $\mathbf{u}$ 's line, exactly when $\theta>90°$ .

Examples

Fixing $\mathbf{u}=\langle 4,0,0\rangle$ and $\mathbf{v}=\langle 2,3,1\rangle$ , the scalar projection is $\mathbf{u}\cdot\mathbf{v}/|\mathbf{u}|=8/4=2$ .