Scalar projection (${\mathrm{comp}}_{\mathbf{c}}\mathbf{a}=\mathbf{a}\cdot \mathbf{c}/\Vert \mathbf{c}\Vert$)

The scalar projection of a real Euclidean vector $\mathbf{a}$ onto a nonzero Euclidean vector $\mathbf{c}$ is its signed component along the unit vector $\mathbf{c}/\Vert \mathbf{c}\Vert$:For $\mathbf{a}=(3,4)$, the scalar projection onto $(2,0)$ is $3$, while that onto $(-2,0)$ is $-3$. Reversing the measuring direction reverses the sign; the scalar projection is not necessarily a nonnegative length. For nonzero $\mathbf{a}$, it also equals $\Vert \mathbf{a}\Vert \cos \theta$, using the angle with $\mathbf{c}$ and its cosine. For the zero vector it is zero, even though that angle is undefined.