Scalar projection
In mathematics, the scalar projection of a vector [math]\displaystyle{ \mathbf{a} }[/math] on (or onto) a vector [math]\displaystyle{ \mathbf{b} }[/math], also known as the scalar resolute of [math]\displaystyle{ \mathbf{a} }[/math] in the direction of [math]\displaystyle{ \mathbf{b} }[/math], is given by:
- [math]\displaystyle{ s = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b}, }[/math]
where the operator [math]\displaystyle{ \cdot }[/math] denotes a dot product, [math]\displaystyle{ \hat{\mathbf{b}} }[/math] is the unit vector in the direction of [math]\displaystyle{ \mathbf{b} }[/math], [math]\displaystyle{ \left\|\mathbf{a}\right\| }[/math] is the length of [math]\displaystyle{ \mathbf{a} }[/math], and [math]\displaystyle{ \theta }[/math] is the angle between [math]\displaystyle{ \mathbf{a} }[/math] and [math]\displaystyle{ \mathbf{b} }[/math].
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 [math]\displaystyle{ \mathbf{a} }[/math] on [math]\displaystyle{ \mathbf{b} }[/math], with a negative sign if the projection has an opposite direction with respect to [math]\displaystyle{ \mathbf{b} }[/math].
Multiplying the scalar projection of [math]\displaystyle{ \mathbf{a} }[/math] on [math]\displaystyle{ \mathbf{b} }[/math] by [math]\displaystyle{ \mathbf{\hat b} }[/math] converts it into the above-mentioned orthogonal projection, also called vector projection of [math]\displaystyle{ \mathbf{a} }[/math] on [math]\displaystyle{ \mathbf{b} }[/math].
Definition based on angle θ
If the angle [math]\displaystyle{ \theta }[/math] between [math]\displaystyle{ \mathbf{a} }[/math] and [math]\displaystyle{ \mathbf{b} }[/math] is known, the scalar projection of [math]\displaystyle{ \mathbf{a} }[/math] on [math]\displaystyle{ \mathbf{b} }[/math] can be computed using
- [math]\displaystyle{ s = \left\|\mathbf{a}\right\| \cos \theta . }[/math] ([math]\displaystyle{ s = \left\|\mathbf{a}_1\right\| }[/math] in the figure)
Definition in terms of a and b
When [math]\displaystyle{ \theta }[/math] is not known, the cosine of [math]\displaystyle{ \theta }[/math] can be computed in terms of [math]\displaystyle{ \mathbf{a} }[/math] and [math]\displaystyle{ \mathbf{b} }[/math], by the following property of the dot product [math]\displaystyle{ \mathbf{a} \cdot \mathbf{b} }[/math]:
- [math]\displaystyle{ \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{a}\right\| \, \left\|\mathbf{b}\right\|} = \cos \theta \, }[/math]
By this property, the definition of the scalar projection [math]\displaystyle{ s \, }[/math] becomes:
- [math]\displaystyle{ 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\| }\, }[/math]
Properties
The scalar projection has a negative sign if [math]\displaystyle{ 90 \lt \theta \le 180 }[/math] 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 [math]\displaystyle{ \mathbf{a}_1 }[/math] and its length [math]\displaystyle{ \left\|\mathbf{a}_1\right\| }[/math]:
- [math]\displaystyle{ s = \left\|\mathbf{a}_1\right\| }[/math] if [math]\displaystyle{ 0 \lt \theta \le 90 }[/math] degrees,
- [math]\displaystyle{ s = -\left\|\mathbf{a}_1\right\| }[/math] if [math]\displaystyle{ 90 \lt \theta \le 180 }[/math] degrees.
See also
- Scalar product
- Cross product
- Vector projection
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