math lessons - Vector resolute

The vector resolute of two vectors, $\mathbf{b}$ in the direction of $\mathbf{a}$ (also " $\mathbf{b}$ on $\mathbf{a}$ "), is given by:

$(\mathbf{b}\cdot\mathbf{\hat a})\mathbf{\hat a}$ or $(|\mathbf{b}|\cos\theta)\mathbf{\hat a}$ (where $θ$ is the angle between vectors $\mathbf{a}$ and $\mathbf{b}$ ).

The vector resolute is a vector, and represents the vector $\mathbf{b}$ "mapped" onto vector $\mathbf{a}$ . The vector resolute is also said to be a component of vector $\mathbf{b}$ in the direction of vector $\mathbf{a}$ .

The other component of $\mathbf{b}$ (perpendicular to $\mathbf{a}$ ) is given by:

$\mathbf{b}\ -\ (\mathbf{b}\cdot\mathbf{\hat a})\mathbf{\hat a}$