Editing Scalar projection

{{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>.

==Definition based on angle ''θ''==
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#Dot product|angle]], ''θ''.

==Definition in terms of a and b==
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>

==Properties==
The scalar projection has a negative sign if <math>90^\circ < \theta \le 180^\circ</math>. It coincides with the [[Euclidean norm|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>

==See also==
* [[Scalar product]]
* [[Cross product]]
* [[Vector projection]]

==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 ==
{{Reflist}}

[[Category:Operations on vectors]]