Scalar projection

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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 <math>\mathbf{a}</math> on (or onto) a vector <math>\mathbf{b},</math> also known as the scalar resolute of <math>\mathbf{a}</math> in the direction of <math>\mathbf{b},</math> is given by:

<math>s = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b},</math>

where the operator <math>\cdot</math> denotes a dot product, <math>\hat{\mathbf{b}}</math> is the unit vector in the direction of <math>\mathbf{b},</math> <math>\left\|\mathbf{a}\right\|</math> is the length of <math>\mathbf{a},</math> and <math>\theta</math> is the angle between <math>\mathbf{a}</math> and <math>\mathbf{b}</math>.<ref>Template:Cite book</ref>

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>\mathbf{a}</math> on <math>\mathbf{b}</math>, with a negative sign if the projection has an opposite direction with respect to <math>\mathbf{b}</math>.

Multiplying the scalar projection of <math>\mathbf{a}</math> on <math>\mathbf{b}</math> by <math>\mathbf{\hat b}</math> converts it into the above-mentioned orthogonal projection, also called vector projection of <math>\mathbf{a}</math> on <math>\mathbf{b}</math>.

Definition based on angle θEdit

If the angle <math>\theta</math> between <math>\mathbf{a}</math> and <math>\mathbf{b}</math> is known, the scalar projection of <math>\mathbf{a}</math> on <math>\mathbf{b}</math> can be computed using

<math>s = \left\|\mathbf{a}\right\| \cos \theta .</math> (<math>s = \left\|\mathbf{a}_1\right\|</math> in the figure)

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

Definition in terms of a and bEdit

When <math>\theta</math> is not known, the cosine of <math>\theta</math> can be computed in terms of <math>\mathbf{a}</math> and <math>\mathbf{b},</math> by the following property of the dot product <math> \mathbf{a} \cdot \mathbf{b}</math>:

<math> \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>s</math> becomes:

<math> 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>

PropertiesEdit

The scalar projection has a negative sign if <math>90^\circ < \theta \le 180^\circ</math>. 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>\mathbf{a}_1</math> and its length <math>\left\|\mathbf{a}_1\right\|</math>:

<math>s = \left\|\mathbf{a}_1\right\| </math> if <math>0^\circ \le \theta \le 90^\circ,</math>

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

SourcesEdit

ReferencesEdit

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