Vector resolute

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:

$(mathbf\{a\}cdotmathbf\{hat\; b\})mathbf\{hat\; b\}$ or $(|mathbf\{a\}|costheta)mathbf\{hat\; b\}$

where $theta$ 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:

$mathbf\{a\}\; -\; (mathbf\{a\}cdotmathbf\{hat\; b\})mathbf\{hat\; b\}$

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:

$|C|\; =\; frac$

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

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

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

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

> = 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.

See also

• Scalar resolute