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{{merge to|Vector projection|date=April 2013}}
 
[[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.png|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''' or '''scalar component''' of <math>\mathbf{a}</math> in the direction of <math>\mathbf{b}</math>, is given by:
 
:<math>s = |\mathbf{a}|\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>|\mathbf{a}|</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>.
 
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 minus 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 = |\mathbf{a}| \cos \theta .</math>
 
==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}} {|\mathbf{a}| \, |\mathbf{b}|} = \cos \theta \,</math>
 
By this property, the definition of the scalar projection <math>s \,</math> becomes:
: <math>s = |\mathbf{a}| \cos \theta = |\mathbf{a}| \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{a}| \, |\mathbf{b}|} = \frac {\mathbf{a} \cdot \mathbf{b}} {|\mathbf{b}| }\,</math>
 
==Properties==
The scalar projection has a negative sign if <math>90 < \theta \le 180</math> degrees. 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>|\mathbf{a}_1|</math>:
 
: <math>s = |\mathbf{a}_1| </math> if <math>0  < \theta \le 90 </math> degrees,
: <math>s = -|\mathbf{a}_1| </math> if <math>90 < \theta \le 180</math> degrees.
 
==See also==
* [[Scalar product]]
* [[Cross product]]
* [[Vector projection]]
 
[[Category:Vectors]]

Revision as of 22:08, 12 September 2013



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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 on (or onto) a vector , also known as the scalar resolute or scalar component of in the direction of , is given by:

where the operator denotes a dot product, is the unit vector in the direction of , is the length of , and is the angle between and .

The scalar projection is a scalar, equal to the length of the orthogonal projection of on , with a minus sign if the projection has an opposite direction with respect to .

Multiplying the scalar projection of on by converts it into the above-mentioned orthogonal projection, also called vector projection of on .

Definition based on angle θ

If the angle between and is known, the scalar projection of on can be computed using

Definition in terms of a and b

When is not known, the cosine of can be computed in terms of and , by the following property of the dot product :

By this property, the definition of the scalar projection becomes:

Properties

The scalar projection has a negative sign if degrees. 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 and its length :

if degrees,
if degrees.

See also