Editing Vector projection (section)

==Definitions in terms of a and b ==
When {{mvar|θ}} is not known, the cosine of {{mvar|θ}} can be computed in terms of {{math|'''a'''}} and {{math|'''b'''}}, by the following property of the [[dot product]] {{math|'''a''' &sdot; '''b'''}}
<math display="block"> \mathbf{a} \cdot \mathbf{b} = \left\|\mathbf{a}\right\| \left\|\mathbf{b}\right\| \cos \theta</math>

===Scalar projection===
By the above-mentioned property of the dot product, the definition of the scalar projection becomes:<ref name=":1" />

In two dimensions, this becomes
<math display="block">a_1 = \frac {\mathbf{a}_x \mathbf{b}_x + \mathbf{a}_y \mathbf{b}_y} {\left\|\mathbf{b}\right\|}.</math>

===Vector projection===
Similarly, the definition of the vector projection of {{math|'''a'''}} onto {{math|'''b'''}} becomes:<ref name=":1" />
<math display="block">\mathbf{a}_1 = a_1 \mathbf{\hat b} = \frac {\mathbf{a} \cdot \mathbf{b}} {\left\|\mathbf{b}\right\| } \frac {\mathbf{b}} {\left\|\mathbf{b}\right\|},</math>
which is equivalent to either
<math display="block">\mathbf{a}_1 = \left(\mathbf{a} \cdot \mathbf{\hat b}\right) \mathbf{\hat b},</math>
or<ref>{{cite web|url=http://www.math.oregonstate.edu/home/programs/undergrad/CalculusQuestStudyGuides/vcalc/dotprod/dotprod.html | title=Dot Products and Projections}} {{Dead link|date=January 2025}} </ref>
<math display="block">\mathbf{a}_1 = \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>

===Scalar rejection===
In two dimensions, the scalar rejection is equivalent to the projection of {{math|'''a'''}} onto <math>\mathbf{b}^\perp = \begin{pmatrix}-\mathbf{b}_y & \mathbf{b}_x\end{pmatrix}</math>, which is <math>\mathbf{b} = \begin{pmatrix}\mathbf{b}_x & \mathbf{b}_y\end{pmatrix}</math> rotated 90° to the left. Hence,
<math display="block">a_2 = \left\|\mathbf{a}\right\| \sin \theta = \frac {\mathbf{a} \cdot \mathbf{b}^\perp} {\left\|\mathbf{b}\right\|} = \frac {\mathbf{a}_y \mathbf{b}_x - \mathbf{a}_x \mathbf{b}_y} {\left\|\mathbf{b}\right\| }.</math>

Such a dot product is called the "perp dot product."

===Vector rejection===
By definition,
<math display="block">\mathbf{a}_2 = \mathbf{a} - \mathbf{a}_1 </math>

Hence,
<math display="block">\mathbf{a}_2 = \mathbf{a} - \frac {\mathbf{a} \cdot \mathbf{b}} {\mathbf{b} \cdot \mathbf{b}}{\mathbf{b}}.</math>

By using the Scalar rejection using the perp dot product this gives

<math display="block">\mathbf{a}_2 = \frac{\mathbf{a}\cdot\mathbf{b}^\perp}{\mathbf{b}\cdot\mathbf{b}}\mathbf{b}^\perp</math>