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 $𝐚$ on (or onto) a vector $𝐛,$ also known as the scalar resolute of $𝐚$ in the direction of $𝐛,$ is given by:

$s=\Vert 𝐚\Vert \cos \theta =𝐚\cdot \widehat{𝐛},$

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

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 $𝐚$ on $𝐛$, with a negative sign if the projection has an opposite direction with respect to $𝐛$.

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

Definition based on angle θ

If the angle $\theta$ between $𝐚$ and $𝐛$ is known, the scalar projection of $𝐚$ on $𝐛$ can be computed using

$s=\Vert 𝐚\Vert \cos \theta .$ ($s=\Vert {𝐚}_1\Vert$ in the figure)

The formula above can be inverted to obtain the angle, θ.

Definition in terms of a and b

When $\theta$ is not known, the cosine of $\theta$ can be computed in terms of $𝐚$ and $𝐛,$ by the following property of the dot product $𝐚\cdot 𝐛$:

$\frac{𝐚\cdot 𝐛}{\Vert 𝐚\Vert \Vert 𝐛\Vert }=\cos \theta$

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

$s=\Vert {𝐚}_1\Vert =\Vert 𝐚\Vert \cos \theta =\Vert 𝐚\Vert \frac{𝐚\cdot 𝐛}{\Vert 𝐚\Vert \Vert 𝐛\Vert }=\frac{𝐚\cdot 𝐛}{\Vert 𝐛\Vert }$

Properties

The scalar projection has a negative sign if $90^{\circ }<\theta \le 180^{\circ }$. 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 ${𝐚}_1$ and its length $\Vert {𝐚}_1\Vert$:

$s=\Vert {𝐚}_1\Vert$ if $0^{\circ }\le \theta \le 90^{\circ },$

$s=-\Vert {𝐚}_1\Vert$ if $90^{\circ }<\theta \le 180^{\circ }.$

See also

• Scalar product
• Cross product
• Vector projection

Sources

• Dot products - www.mit.org
• Scalar projection - Flexbooks.ck12.org
• Scalar Projection & Vector Projection - medium.com
• Lesson Explainer: Scalar Projection | Nagwa