Scalar (or Dot) product: Formulas, Properties & Finding angle between 2 vectors

Formula

$$ \boxed{ \textbf{a} \cdot \textbf{b} = | \textbf{a} | | \textbf{b} | \cos \theta } $$

If vector $\textbf{a} = \left( \begin{matrix} a_1 \\ a_2 \\ a_3 \end{matrix} \right) $ and vector $\textbf{b} = \left( \begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right) $, $$ \boxed{ \textbf{a} \cdot \textbf{b} = \left( \begin{matrix} a_1 \\ a_2 \\ a_3 \end{matrix} \right) \cdot \left( \begin{matrix} b_1 \\ b_2 \\ b_3 \end{matrix} \right) = a_1 b_1 + a_2 b_2 + a_3 b_3 } $$

Angle between two vectors

From $ \textbf{a} \cdot \textbf{b} = | \textbf{a} | | \textbf{b} | \cos \theta$, $$ \boxed{ \cos \theta = { \textbf{a} \cdot \textbf{b} \over | \textbf{a} | |\textbf{b} | } } $$

If $ \textbf{a} \cdot \textbf{b} $ is a positive value, the angle between the two vectors is acute:

If $ \textbf{a} \cdot \textbf{b} $ is a negative value, the angle between the two vectors is obtuse:

Perpendicular vectors

If vectors $\textbf{a}$ and $\textbf{b}$ are perpendicular, $$ \boxed{ \textbf{a} \cdot \textbf{b} = 0 } $$

Recall that since $\cos 90^\circ = 0$, \begin{align} \textbf{a} \cdot \textbf{b} & = | \textbf{a} | |\textbf{b} | \cos 90^\circ \\ & = | \textbf{a} | | \textbf{b} |(0) \\ & = 0 \end{align}

Properties

Commutative property: $$ \boxed{\textbf{a} \cdot \textbf{b} = \textbf{b} \cdot \textbf{a}} $$

Distributive property: $$ \boxed{ \textbf{a} \cdot (\textbf{b} + \textbf{c}) = \textbf{a} \cdot \textbf{b} + \textbf{a} \cdot \textbf{c} }$$ $$ \boxed{ (\textbf{b} + \textbf{c}) \cdot \textbf{a} = \textbf{b} \cdot \textbf{a} + \textbf{c} \cdot \textbf{a} } $$

Associative property: If $\lambda$ is a constant, $$ \boxed{ \textbf{a} \cdot (\lambda \textbf{b}) = (\lambda \textbf{a}) \cdot \textbf{b} = \lambda (\textbf{a} \cdot \textbf{b} ) } $$

Value of $\textbf{a} \cdot \textbf{a}$: $$ \boxed{ \textbf{a} \cdot \textbf{a} = | \textbf{a} | ^2 }$$

The last property can be used for any vectors. If we consider $\textbf{b} + \textbf{c}$ as a vector, $$ (\textbf{b} + \textbf{c}) \cdot (\textbf{b} + \textbf{c}) = | \textbf{b} + \textbf{c} |^2 $$