Vector_resolute

Vector resolute

It has been suggested that this article or section be merged with scalar resolute. (Discuss)

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 $\mathbf{a}$ 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).

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 $B$ :

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

giving the final formula:

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

Uses

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

Scalar · Vector · Vector space · Vector projection · Linear span · Linear map · Linear projection · Linear independence · Linear combination · Basis · Column space · Row space · Dual space · Orthogonality · Rank · Minor · Kernel (matrix) · Eigenvalue, eigenvector and eigenspace · Least squares regressions · Outer product · Inner product space · Cross product · Dot product · Transpose · Gram–Schmidt process · Matrix decomposition

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Vector resolute overview

Uses

See also