Scalar projection

File:Dot Product.svg

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

File:Projection and rejection.png

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:

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

where the operator $\cdot$ denotes a dot product, $\hat{\mathbf{b}}$ is the unit vector in the direction of $\mathbf{b},$ $\Vert \mathbf{a}\Vert$ 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 $\hat{\mathbf{b}}$ 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

$s=\Vert \mathbf{a}\Vert \cos \theta .$ ($s=\Vert {\mathbf{a}}_1\Vert$ 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 $\mathbf{a}$ and $\mathbf{b},$ by the following property of the dot product $\mathbf{a}\cdot \mathbf{b}$:

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

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

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

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 ${\mathbf{a}}_1$ and its length $\Vert {\mathbf{a}}_1\Vert$:

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

$s=-\Vert {\mathbf{a}}_1\Vert$ 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