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 $(|mathbf\{a\}|cos\; heta)mathbf\{hat\; b\}$

where $heta$ 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:

$|C|\; =\; |A|\; cos\; heta$

To calculate $C$ use the definition of the dot product:$A\; cdot\; B\; =\; |A|\; ,\; |B|\; cos\; heta\; ,$

Using the above equation:

$|C|\; =\; |A|\; cos\; heta$

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

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

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.

ee also

*Scalar resolute