10.3 Dot Product and Projection of a Vector On a Line > No. 4

Find the projection of one vector on another vector.

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Explanation:

Dot product

In mathematics, the dot product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number obtained by multiplying corresponding entries and then summing those products. The name is derived from the centered dot "·" that is often used to designate this operation; the alternative name scalar product emphasizes the scalar (rather than vector) nature of the result. At a basic level, the dot product is used to obtain the cosine of the angle between two vectors.

Geometric interpretation

$Description: \mathbf{A}_B = \left\|\mathbf{A}\right\| \cos\theta$ is the scalar projection of $Description: \mathbf{A}$ onto $Description: \mathbf{B}$ .
Since $Description: \mathbf{A} \cdot \mathbf{B} = \left\|\mathbf{A}\right\| \left\|\mathbf{B}\right\| \cos\theta$ , then $Description: \mathbf{A}_B = \frac{\mathbf{A} \cdot \mathbf{B}}{\left\|\mathbf{B}\right\|}$ .

In Euclidean geometry, the dot product of vectors expressed in an orthonormal basis is related to their length and angle. For such a vector $Description: \mathbf{a}$ , the dot product $Description: \mathbf{a}\cdot\mathbf{a}$ is the square of the length of $Description: \mathbf{a}$ , or

$Description: {\mathbf{a} \cdot \mathbf{a}}=\left\|\mathbf{a}\right\|^2$

where $Description: \left\|\mathbf{a}\right\|$ denotes the length (magnitude) of $Description: \mathbf{a}$ . If $Description: \mathbf{b}$ is another such vector,

$Description: \mathbf{a} \cdot \mathbf{b}=\left\|\mathbf{a}\right\| \, \left\|\mathbf{b}\right\| \cos \theta \,$

where $θ$ is the angle between them.

This formula can be rearranged to determine the size of the angle between two nonzero vectors:

$Description: \theta=\arccos \left( \frac {\bold{a}\cdot\bold{b}} {\left\|\bold{a}\right\|\left\|\bold{b}\right\|}\right)$

(Our solved example in mathguru.com uses this concept)

http://en.wikipedia.org/wiki/Dot_product

Vector projection

The vector projection (also known as the vector resolute, or vector component) of a vector $Description: \mathbf{a}$ in the direction of a vector $Description: \mathbf{b}$ (or "of $Description: \mathbf{a}$ on/onto $Description: \mathbf{b}$ "), is given by:

$Description: (\mathbf{a}\cdot\mathbf{\hat b})\mathbf{\hat b} = (|\mathbf{a}|\cos\theta)\mathbf{\hat b}$

where $θ$ is the angle between the vectors $Description: \mathbf{b}$ and $Description: \mathbf{a}$ ; the operator $Description: \cdot$ is the dot product; and $Description: \hat{\mathbf{b}}$ is the unit vector in the direction of $Description: \mathbf{b}$ .

(Our solved example in mathguru.com uses this concept)

The other component of $Description: \mathbf{a}$ (perpendicular to $Description: \mathbf{b}$ ), called the vector rejection of $Description: \mathbf{a}$ from $Description: \mathbf{b}$ , is given by:

$Description: \mathbf{a}\ -\ (\mathbf{a}\cdot\mathbf{\hat b})\mathbf{\hat b}.$

Both the vector projection and the vector rejection are vectors. The vector projection of $Description: \mathbf{a}$ on $Description: \mathbf{b}$ is the orthogonal projection of $Description: \mathbf{a}$ onto the straight line defined by $Description: \mathbf{b}$ . The corresponding vector rejection is the orthogonal projection of $Description: \mathbf{a}$ onto a plane orthogonal to $Description: \mathbf{b}$ .

The vector projection of $Description: \mathbf{a}$ on $Description: \mathbf{b}$ can be also regarded as the corresponding scalar projection $Description: (\mathbf{a}\cdot\mathbf{\hat b})$ multiplied by $Description: \mathbf{\hat b}$ .

Overview

If $A$ and $B$ are two vectors, the projection of $A$ on $B$ is the vector $C$ with the same direction as $B$ and with the length:

$Description: |C| = |A| \cos \theta\,$

When $θ$ is not known, we can compute $Description: \cos \theta \,$ using the following property of the dot product $Description: A \cdot B$ :

$Description: \frac {A \cdot B} {|A| \, |B|} = \cos \theta \,$

Thus, the length of C can be also computed as follows

$Description: |C| = |A| \cos \theta = |A| \frac {A \cdot B} {|A| \, |B|} = \frac {A \cdot B} {|B| }\,$

Since $C$ is in the same direction as $B$ ,

$Description: C = |C| {\hat B}$

where $Description: {\hat B}$ is the unit vector with the same direction as $B$ :

$Description: {\hat B} = \frac {B} {|B|}\,$

Substituting, we obtain

$Description: C = \frac {A \cdot B} {|B|} {\hat B},$

which is equivalent to either

$Description: C = (A \cdot \hat B) {\hat B},$

$Description: C = \frac {A \cdot B} {|B| } \frac {B} {|B|} = \frac {A \cdot B} {|B|^2}{B} = \frac {A \cdot B} {B \cdot B}{B}.$

The latter formula is computationally more efficient than the former, as the former requires three multiplications and a square root to compute |B| (in turn, needed to compute $Description: {\hat B}$ ), while the latter only requires three multiplications to compute $Description: {B \cdot B}$ (the remaining parts of the two formulas require the same number and kind of basic algebraic operations).

http://en.wikipedia.org/wiki/Vector_projection

The above explanation is copied from Wikipedia, the free encyclopedia and is remixed as allowed under the Creative Commons Attribution- ShareAlike 3.0 Unported License.