Scalar projection

If 0° ≤ θ ≤ 90°, as in this case, the scalar projection of a on b coincides with the length of the vector projection.

Vector projection of a on b (a₁), and vector rejection of a from b (a₂).

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:

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

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.

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

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

$s = |\mathbf{a}| \cos \theta .$

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 $\mathbf{a} \cdot \mathbf{b}$:

$\frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{a}| \, |\mathbf{b}|} = \cos \theta \,$

By this property, the definition of the scalar projection $s \,$ becomes:

$s = |\mathbf{a}| \cos \theta = |\mathbf{a}| \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{a}| \, |\mathbf{b}|} = \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{b}| }\,$

Properties

The scalar projection has a negative sign if $90 < \theta \le 180$ degrees. 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 $|\mathbf{a}_1|$:

$s = |\mathbf{a}_1|$ if $0 < \theta \le 90$ degrees,

$s = -|\mathbf{a}_1|$ if $90 < \theta \le 180$ degrees.

See also

• Scalar product
• Cross product
• Vector projection