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

where $theta$ 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 $costheta\; =\; frac$

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.

See also

• vector resolute
• scalar product
• cross product