Given two vectors \(\bm{u} = \langle a_1,b_1 \rangle\) and \(\bm{v} = \langle a_2,b_2 \rangle,\) their dot product \(\bm{u}\cdot\bm{v}\) can be calculated as \(\bm{u}\cdot\bm{v} = a_1a_2+b_1b_2.\) Alternatively, if \(\theta\) is the angle between \(\bm{u}\) and \(\bm{v},\) \(\bm{u}\cdot\bm{v} = |\bm{u}||\bm{v}|\cos(\theta).\)
If two vectors are perpendicular (orthogonal) if \(\bm{u}\cdot\bm{v} =0,\) and two vectors are parallel if \(\bm{u}\cdot\bm{v} = |\bm{u}||\bm{v}|.\)
The component of \(\bm{u}\) along \(\bm{v}\) is either \(\bm{u}\cos(\theta)\) or equivalently \(\frac{\bm{u}\cdot\bm{v}}{|\bm{v}|}.\)
The projection of \(\bm{u}\) onto \(\bm{v}\), denoted \(\operatorname{proj}_{\bm{v}}(\bm{u}),\) is \[(\text{the component of }\(\bm{u}\) \text{ along } \(\bm{v})\times(\term{the unit vector in the direction of } \bm{v} ) = \frac{\bm{u}\cdot\bm{v}}{|\bm{v}|^2}\bm{v} \]
We can resolve a vector \(\bm{u}\) by writing it as a sum of orthogonal vectors, \[ \bm{u} = \Big(\operatorname{proj}_{\bm{v}}(\bm{u})\Big) + \Big(\bm{u} - \operatorname{proj}_{\bm{v}}(\bm{u})\Big) \]