Scalar projection

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

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

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

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

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