Editing Scalar projection (section)

{{Short description|Mathematics visualization}}
{{more citations needed|date=March 2024}}
[[Image:Dot Product.svg|thumb|300px|right|If 0° ≤ ''θ'' ≤ 90°, as in this case, the scalar projection of '''a''' on '''b''' coincides with the [[Euclidean norm|length]] of the [[vector projection]].]]
[[File:Projection and rejection.svg|thumb|200px|[[Vector projection]] of '''a''' on '''b''' ('''a'''<sub>1</sub>), and vector rejection of '''a''' from '''b''' ('''a'''<sub>2</sub>).]]

In [[mathematics]], the '''scalar projection''' of a [[vector (geometric)|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 (geometry)|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 [[Euclidean norm|length]] of <math>\mathbf{a},</math> and <math>\theta</math> is the [[angle]] between <math>\mathbf{a}</math> and <math>\mathbf{b}</math>.<ref>{{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}}</ref>

The term '''scalar component''' refers sometimes to scalar projection, as, in [[Cartesian coordinates]], the [[Basis (linear algebra)|components of a vector]] are the scalar projections in the directions of the [[coordinate axes]].

The scalar projection is a [[scalar (mathematics)|scalar]], equal to the [[Euclidean norm|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>.