scalar projection

In mathematics, the scalar projection of a vector $\mathbf$ on (or onto) a vector $\mathbf,$ also known as the scalar resolute of $\mathbf$ in the direction of $\mathbf,$ is given by:

:$s = \left\, \mathbf\right\, \cos\theta = \mathbf\cdot\mathbf,$

where the operator $\cdot$ denotes a dot product, $\hat$ is the unit vector in the direction of $\mathbf,$ $\left\, \mathbf\right\,$ is the length of $\mathbf,$ and $\theta$ is the angle between $\mathbf$ and $\mathbf$.

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

Multiplying the scalar projection of $\mathbf$ on $\mathbf$ by $\mathbf$ converts it into the above-mentioned orthogonal projection, also called vector projection of $\mathbf$ on $\mathbf$.

Definition based on angle ''θ''

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

:$s = \left\, \mathbf\right\, \cos \theta .$ ($s = \left\, \mathbf_1\right\,$ 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 $\mathbf$ and $\mathbf,$ by the following property of the dot product $\mathbf \cdot \mathbf$:
: $\frac = \cos \theta$

By this property, the definition of the scalar projection $s$ becomes:
: $s = \left\, \mathbf_1\right\, = \left\, \mathbf\right\, \cos \theta = \left\, \mathbf\right\, \frac = \frac \,$

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 $\mathbf_1$ and its length $\left\, \mathbf_1\right\,$:

: $s = \left\, \mathbf_1\right\,$ if $0^\circ \le \theta \le 90^\circ,$
: $s = -\left\, \mathbf_1\right\,$ if $90^\circ < \theta \le 180^\circ.$

See also

* Scalar product
* Cross product
* Vector projection

Sources

Dot products - www.mit.orgScalar projection - Flexbooks.ck12.orgScalar Projection & Vector Projection - medium.comLesson Explainer: Scalar Projection , Nagwa

References

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Operations on vectors