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Vector projection: Wikis


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The vector resolute (also known as the vector projection) of two vectors, \mathbf{a} in the direction of \mathbf{b} (also "\mathbf{a} on \mathbf{b}"), is given by:

(\mathbf{a}\cdot\mathbf{\hat b})\mathbf{\hat b}\text{ or }(|\mathbf{a}|\cos\theta)\mathbf{\hat b}

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

The vector resolute is a vector, and is the orthogonal projection of the vector \mathbf{a} onto the vector \mathbf{b}. The vector resolute is also said to be a component of vector \mathbf{a} in the direction of vector \mathbf{b}.

The other component of \mathbf{a} (perpendicular to \mathbf{b}) is given by:(By triangle addition of vectors)

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

The vector resolute is also the scalar resolute multiplied by \mathbf{\hat b} (in order to convert it into a vector, or give it direction).e.g. in this case scalar resolute is (\mathbf{a}\cdot\mathbf{\hat b}) .


Vector resolute overview

Dot Product.svg

If A and B are two vectors, the projection (C) of A on B is the vector that has the same slope as B with the length:

|C| = |A| \cos \theta\,

To calculate C use the following property of the dot product:  A \cdot B = |A| \, |B| \cos \theta \,

Using the above equation:

|C| = |A| \cos \theta\,

Multiply and divide by | B | at the same time:

|C| = \frac {|A| |B| \cos \theta} {|B| }\,

In the resulting fraction, the top term is the same as the dot product, hence:

|C| = \frac {A \cdot B} {|B| }\,

To find the length of | C | with an unknown θ, and unknown direction, multiply it with the unit vector B:

C = \frac {A \cdot B} {|B| } \frac {B} {|B|} = \frac {A \cdot B} {B \cdot B} B,

giving the final formula:

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

Matrix representation

The orthogonal projection can be represented by a projection matrix. To project a vector onto the unit vector a = (ax, ay, az), it would need to be multiplied with this projection matrix:

 P_a = a a^T = \begin{bmatrix} a_1 \\ a_2 \\ a_3 \end{bmatrix} \begin{bmatrix} a_1 & a_2 & a_3 \end{bmatrix} = \begin{bmatrix} a_1^2 & a_1 a_2 & a_1 a_3 \ a_1 a_2 & a_2^2 & a_2 a_3 \ a_1 a_3 & a_2 a_3 & a_3^2 \ \end{bmatrix}.

Vector projection

To project a certain vector onto an other vector, the length of the projection on this new vector can be calculated by the vector product of these two vectors.

eg. When a 3 dimensional vector eg. A=[3 4 5] has to be projected onto an other 3 dimensional vector eg. B=[6 7 8], the size of the projection is equal to the scalar matrix product A * B'. = 3 * 6 + 4 * 7 + 5 * 8 = 86.This is the length of the projection on the vector B. To express this as a vector we have to multiply this result with the unity vector of Eb which is equal to B/|B|=[6 7 8]/12.2 =[0.4915 0.5735 0.6554]. So the projection vector is [0.4915 0.5735 0.6554] * 86=[42.2690 49.3210 56.3644]


The vector projection is an important operation in the Gram-Schmidt orthonormalization of vector space bases.

See also



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