Properties of Dot Product

The dot product obeys many of the laws that hold for ordinary products of real numbers. These are stated in the following theorem.

Properties of the Dot Product 2 If a, b, and c are vectors in \(V_3\) and \(c\) is a scalar, then 1. \(\mathbf{a} \cdot \mathbf{a} = |\mathbf{a}|^2\) 2. \(\mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a}\) 3. \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}\) 4. \((c\mathbf{a}) \cdot \mathbf{b} = c(\mathbf{a} \cdot \mathbf{b}) = \mathbf{a} \cdot (c\mathbf{b})\) 5. \(\mathbf{0} \cdot \mathbf{a} = 0\)

These properties are easily proved using Definition 1. For instance, here are the proofs of Properties 1 and 3: 1. \(\mathbf{a} \cdot \mathbf{a} = a_1^2 + a_2^2 + a_3^2 = |\mathbf{a}|^2\) 3. \(\mathbf{a} \cdot (\mathbf{b} + \mathbf{c}) = \langle a_1, a_2, a_3 \rangle \cdot \langle b_1 + c_1, b_2 + c_2, b_3 + c_3 \rangle\) \(= a_1(b_1 + c_1) + a_2(b_2 + c_2) + a_3(b_3 + c_3)\) \(= a_1b_1 + a_1c_1 + a_2b_2 + a_2c_2 + a_3b_3 + a_3c_3\) \(= (a_1b_1 + a_2b_2 + a_3b_3) + (a_1c_1 + a_2c_2 + a_3c_3)\) \(= \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c}\)

The proofs of the remaining properties are left as exercises.