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:

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

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 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

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

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 $90<\theta \leq 180$ 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 $\mathbf {a} _{1}$ and its length $|{\mathbf {a}}_{1}|$:

$s=|{\mathbf {a}}_{1}|$ if $0<\theta \leq 90$ degrees,

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

See also

• Scalar product
• Cross product
• Vector projection