Vector resolute

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

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

where $θ$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ and $\hat{\mathbf{a}}$ is the unit vector in the direction of $\mathbf{a}$ .

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

The other component of $\mathbf{b}$ (perpendicular to $\mathbf{a}$ ) is given by:

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

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

[edit] Vector resolute overview

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

$|C| = |A| \cdot \cos \theta$

To calculate  $C$  we need the definition of the dotproduct:

Let's start:

$|C| = |A| \cdot \cos \theta$

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

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

Recognize the top term? Same as the dotproduct, so we get this:

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

Great, now we can get the length of $| C |$ without knowing $θ$ But we still don't have the direction of it, we have to multiply it with the normalized vector $B$ :

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