scalar projection

Image:Dot Product.svg of the vector projection.]]

File:Projection and rejection.svg 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:

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

where the operator $\cdot$ denotes a dot product, $\hat{\mathbf{b}}$ is the unit vector in the direction of $\mathbf{b},$ $\left\|\mathbf{a}\right\|$ is the length of $\mathbf{a},$ and $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$ .{{Cite book |last=Strang |first=Gilbert |title=Introduction to linear algebra |date=2016 |publisher=Cambridge press |isbn=978-0-9802327-7-6 |edition=5th |location=Wellesley}}

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

: $s = \left\|\mathbf{a}\right\| \cos \theta .$ ( $s = \left\|\mathbf{a}_1\right\|$ 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 $\mathbf{a} \cdot \mathbf{b}$ :

: $\frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \cos \theta$

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

: $s = \left\|\mathbf{a}_1\right\| = \left\|\mathbf{a}\right\| \cos \theta = \left\|\mathbf{a}\right\| \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| }\,$

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 $\left\|\mathbf{a}_1\right\|$ :

: $s = \left\|\mathbf{a}_1\right\|$ if $0^\circ \le \theta \le 90^\circ,$

: $s = -\left\|\mathbf{a}_1\right\|$ if $90^\circ < \theta \le 180^\circ.$

Sources

[http://www.mit.edu/~hlb/StantonGrant/18.02/details/tex/lec1snip2-dotprod.pdf Dot products - www.mit.org]
[https://flexbooks.ck12.org/cbook/ck-12-college-precalculus/section/9.6/primary/lesson/scalar-and-vector-projections-c-precalc#:~:text=The%20definition%20of%20scalar%20projection%20is%20the%20length%20of%20the%20vector%20projection.&text=A%20scalar%20projection%20is%20given,is%20less%20than%2090%E2%88%98. Scalar projection - Flexbooks.ck12.org]
[https://medium.com/linear-algebra-basics/scalar-projection-vector-projection-5076d89ed8a8 Scalar Projection & Vector Projection - medium.com]
[https://www.nagwa.com/en/explainers/792181370490/ Lesson Explainer: Scalar Projection | Nagwa]

References

Category:Operations on vectors