The dot product (or scalar product) of the two real vectors \[\mathbf{a} = \begin{pmatrix}a_1 \\ a_2 \\ a_3\end{pmatrix} \quad\text{and}\quad \mathbf{b} = \begin{pmatrix}b_1 \\ b_2 \\ b_3 \end{pmatrix}\] is \[\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3.\] This is a real number (hence the name ‘scalar’ product). The definition extends to vectors \(\mathbf{a}\) and \(\mathbf{b}\) with \(n\) components. (Compare this to the cross product, which outputs a vector and which does not have such a simple generalisation.)
The dot product combines information about the lengths of \(\mathbf{a}\) and \(\mathbf{b}\) and the angle between them. Specifically, \[\mathbf{a} \cdot \mathbf{b} = |\mathbf{a}|\, |\mathbf{b}| \cos \theta\] where \(|\mathbf{a}|\) and \(|\mathbf{b}|\) are the lengths (magnitudes) of \(\mathbf{a}\) and \(\mathbf{b}\) and \(\theta\) is the angle between the two vectors. It follows that for non-zero vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product is zero exactly when \(\mathbf{a}\) and \(\mathbf{b}\) are at right angles to one another.
For every choice of \(\mathbf{a}\), \(\mathbf{b}\) and \(\mathbf{c}\) and every scalar \(\lambda\), we have \[\begin{align*} \mathbf{a} \cdot \mathbf{b} &= \mathbf{b} \cdot \mathbf{a} && \text{(commutativity)}\\ \mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) &= \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} && \text{(distributivity over addition)}\\ (\lambda \mathbf{a}) \cdot \mathbf{b} &= \lambda (\mathbf{a} \cdot \mathbf{b}) = \mathbf{a} \cdot (\lambda \mathbf{b}). \end{align*}\]
