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

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

The scalar projection is a scalar, equal to the length of the orthogonal projection of ${\displaystyle \mathbf {a} }$ on ${\displaystyle {\mathbf {b} }}$, with a minus 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 ${\displaystyle {\mathbf {b} }}$.

## Definition based on angle θ

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

## Definition in terms of a and b

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

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

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

## Properties

The scalar projection has a negative sign if ${\displaystyle 90<\theta \leq 180}$ 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 ${\displaystyle {\mathbf {a} }_{1}}$ and its length ${\displaystyle |{\mathbf {a} }_{1}|}$:

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