# Vector projection: Wikis

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# Encyclopedia

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

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]

## Uses

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