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 ⋅ {\displaystyle \cdot } \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 $\mathbf {b}$ .

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

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 a {\displaystyle \mathbf {a} } \mathbf {a} on b {\displaystyle \mathbf {b} } \mathbf {b} , with a negative sign if the projection has an opposite direction with respect to b {\displaystyle \mathbf {b} } \mathbf {b} .

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

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

$s=\left\|\mathbf {a} _{1}\right\|} {\displaystyle s=\left\|\mathbf {a} _{1}\right\|$ in the figure)

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 < θ ≤ 180 {\displaystyle 90<\theta \leq 180} 90<\theta \leq 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 a 1 {\displaystyle \mathbf {a} _{1}} \mathbf {a} _{1} and its length ‖ a 1 ‖ {\displaystyle \left\|\mathbf {a} _{1}\right\|} {\displaystyle \left\|\mathbf {a} _{1}\right\|}:

$s=\left\|\mathbf {a} _{1}\right\|$ if $0<\theta \leq 90$ degrees,
$s=-\left\|\mathbf {a} _{1}\right\|$ if $90<\theta \leq 180$ degrees.