Vector projection

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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:

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

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).

[edit] 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 definition 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:

$C = \frac {A \cdot B} {B \cdot B} B.$

[edit] Matrix representation

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

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