Scalar resolute

The scalar resolute, also known as the scalar projection or scalar component, of a vector $mathbf\{b\}$ in the direction of a vector $mathbf\{a\}$ is given by:

:$mathbf\{b\}cdotmathbf\{hat\; a\}\; =\; |mathbf\{b\}|cos\; heta$

where $heta$ is the angle between the vectors $mathbf\{a\}$ and $mathbf\{b\}$ and $hat\{mathbf\{a$ is the unit vector in the direction of $mathbf\{a\}$. This is also known as "$mathbf\{b\}$ on $mathbf\{a\}$".

For an intuitive understanding of this formula, recall from trigonometry that $cos\; heta\; =\; frac$ and simply rearrange the terms by multiplying both sides by $|mathbf\{b\}|$.

The scalar resolute is a scalar, and is the length of the orthogonal projection of the vector $mathbf\{b\}$ onto the vector $mathbf\{a\}$, with a minus sign if the direction is opposite.

Multiplying the scalar resolute by $mathbf\{hat\; a\}$ converts it into the vector resolute, a vector.

ee also

* vector resolute
* scalar product
* cross product