Scalar projection



In mathematics, the scalar projection of a vector $$\mathbf{a}$$ on (or onto) a vector $$\mathbf{b},$$ also known as the scalar resolute of $$\mathbf{a}$$ in the direction of $$\mathbf{b},$$ is given by:


 * $$s = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b},$$

where the operator $$\cdot$$ denotes a dot product, $$\hat{\mathbf{b}}$$ is the unit vector in the direction of $$\mathbf{b},$$ $$\left\|\mathbf{a}\right\|$$ is the length of $$\mathbf{a},$$ and $$\theta$$ is the angle between $$\mathbf{a}$$ and $$\mathbf{b}$$.

The term scalar component refers sometimes to scalar projection, as, in Cartesian coordinates, the components of a vector are the scalar projections in the directions of the coordinate axes.

The scalar projection is a scalar, equal to the length of the orthogonal projection of $$\mathbf{a}$$ on $$\mathbf{b}$$, with a negative sign if the projection has an opposite direction with respect to $$\mathbf{b}$$.

Multiplying the scalar projection of $$\mathbf{a}$$ on $$\mathbf{b}$$ by $$\mathbf{\hat b}$$ converts it into the above-mentioned orthogonal projection, also called vector projection of $$\mathbf{a}$$ on $$\mathbf{b}$$.

Definition based on angle θ
If the angle $$\theta$$ between $$\mathbf{a}$$ and $$\mathbf{b}$$ is known, the scalar projection of $$\mathbf{a}$$ on $$\mathbf{b}$$ can be computed using


 * $$s = \left\|\mathbf{a}\right\| \cos \theta .$$  ($$s = \left\|\mathbf{a}_1\right\|$$ in the figure)

The formula above can be inverted to obtain the angle, θ.

Definition in terms of a and b
When $$\theta$$ is not known, the cosine of $$\theta$$ can be computed in terms of $$\mathbf{a}$$ and $$\mathbf{b},$$ by the following property of the dot product $$ \mathbf{a} \cdot \mathbf{b}$$:
 * $$ \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta$$

By this property, the definition of the scalar projection $$s$$ becomes:
 * $$ s = \left\|\mathbf{a}_1\right\| = \left\|\mathbf{a}\right\| \cos \theta = \left\|\mathbf{a}\right\| \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| }\,$$

Properties
The scalar projection has a negative sign if $$90^\circ < \theta \le 180^\circ$$. It coincides with the length of the corresponding vector projection if the angle is smaller than 90°. More exactly, if the vector projection is denoted $$\mathbf{a}_1$$ and its length $$\left\|\mathbf{a}_1\right\|$$:


 * $$s = \left\|\mathbf{a}_1\right\| $$ if $$0^\circ  \le \theta \le 90^\circ,$$
 * $$s = -\left\|\mathbf{a}_1\right\| $$ if $$90^\circ < \theta \le 180^\circ.$$