Definitions

# Scalar resolute

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

$mathbf\left\{b\right\}cdotmathbf\left\{hat a\right\} = |mathbf\left\{b\right\}|costheta$

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

For an intuitive understanding of this formula, recall from trigonometry that $costheta = frac$

{|mathbf{b}> and simply rearrange the terms by multiplying both sides by $|mathbf\left\{b\right\}|$.

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

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