Editing Vector projection (section)

{{Short description|Concept in linear algebra}}
{{For|more general concepts|Projection (linear algebra)|Projection (mathematics)}}
{{Lead too long|date=September 2024}}
The '''vector projection''' (also known as the '''vector component''' or '''vector resolution''') of a [[Vector (geometry)|vector]] {{math|'''a'''}} on (or onto) a nonzero vector {{math|'''b'''}} is the [[orthogonal projection]] of {{math|'''a'''}} onto a [[straight line]] parallel to {{math|'''b'''}}. 
The projection of {{math|'''a'''}} onto {{math|'''b'''}} is often written as <math>\operatorname{proj}_\mathbf{b} \mathbf{a}</math> or {{math|'''a'''<sub>∥'''b'''</sub>}}.

The vector component or vector resolute of {{math|'''a'''}} [[perpendicular]] to {{math|'''b'''}}, sometimes also called the '''vector rejection''' of {{math|'''a'''}} ''from'' {{math|'''b'''}} (denoted <math>\operatorname{oproj}_{\mathbf{b}} \mathbf{a}</math> or {{math|'''a'''<sub>⊥'''b'''</sub>}}),<ref>{{cite book |first=G. |last=Perwass |year=2009 |url=https://books.google.com/books?id=8IOypFqEkPMC&pg=PA83 |title=Geometric Algebra With Applications in Engineering |page=83 |publisher=Springer |isbn=9783540890676 }}</ref> is the orthogonal projection of {{math|'''a'''}} onto the [[plane (geometry)|plane]] (or, in general, [[hyperplane]]) that is [[orthogonal]] to {{math|'''b'''}}. Since both <math>\operatorname{proj}_{\mathbf{b}} \mathbf{a}</math> and <math>\operatorname{oproj}_{\mathbf{b}} \mathbf{a}</math> are vectors, and their sum is equal to {{math|'''a'''}}, the rejection of {{math|'''a'''}} from {{math|'''b'''}} is given by: <math display="block">\operatorname{oproj}_{\mathbf{b}} \mathbf{a} = \mathbf{a} - \operatorname{proj}_{\mathbf{b}} \mathbf{a}.</math>

[[File:Projection and rejection.svg|thumb|488x488px|Projection of {{math|'''a'''}} on {{math|'''b'''}} ('''a'''<sub>1</sub>), and rejection of {{math|'''a'''}} from {{math|'''b'''}} ('''a'''<sub>2</sub>).|center]]
[[File:Projection and rejection 2.svg|thumb|248px|When {{math|90° < ''θ'' ≤ 180°}}, {{math|'''a'''<sub>1</sub>}} has an opposite direction with respect to {{math|'''b'''}}.]]
To simplify notation, this article defines <math>\mathbf{a}_1 := \operatorname{proj}_{\mathbf{b}} \mathbf{a}</math> and <math>\mathbf{a}_2 := \operatorname{oproj}_{\mathbf{b}} \mathbf{a}.</math>
Thus, the vector <math>\mathbf{a}_1</math> is parallel to <math>\mathbf{b},</math> the vector <math>\mathbf{a}_2</math> is orthogonal to <math>\mathbf{b},</math> and <math>\mathbf{a} = \mathbf{a}_1 + \mathbf{a}_2.</math> 

The projection of {{math|'''a'''}} onto {{math|'''b'''}} can be decomposed into a direction and a scalar magnitude by writing it as <math>\mathbf{a}_1 = a_1\mathbf{\hat b}</math>
where <math>a_1</math> is a scalar, called the ''[[scalar projection]]'' of {{math|'''a'''}} onto {{math|'''b'''}}, and {{math|'''b̂'''}} is the [[unit vector]] in the direction of {{math|'''b'''}}. The scalar projection is defined as<ref name=":1">{{Cite web|title=Scalar and Vector Projections| url=https://www.ck12.org/book/ck-12-college-precalculus/section/9.6/|access-date=2020-09-07|website=www.ck12.org}}</ref>
<math display="block">a_1 = \left\|\mathbf{a}\right\|\cos\theta = \mathbf{a}\cdot\mathbf{\hat b}</math>
where the operator '''&sdot;''' denotes a [[dot product]], ‖'''a'''‖ is the [[Euclidean norm|length]] of {{math|'''a'''}}, and ''θ'' is the [[angle]] between {{math|'''a'''}} and {{math|'''b'''}}.
The scalar projection is equal in absolute value to the length of the vector projection, with a minus sign if the direction of the projection is [[opposite vector|opposite]] to the direction of {{math|'''b'''}}, that is, if the angle between the vectors is more than 90 degrees.

The vector projection can be calculated using the dot product of <math>\mathbf{a}</math> and <math>\mathbf{b}</math> as:
<math display="block">\operatorname{proj}_{\mathbf{b}} \mathbf{a} = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\|^2}{\mathbf{b}} = \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}} ~ .</math>