13.3 | The Dot Product


The dot product or scalar product of two nonzero vectors \(\mathbf{a}\) and \(\mathbf{b}\) is the number
$$
\mathbf{a \cdot b} = \left| \mathbf{a} \right|\left| \mathbf{b} \right| \cos \theta
$$
where \( \theta \) is the angle between \( \mathbf{a} \) and \( \mathbf{b} \), \(0 \leq \theta \leq \pi \). (So \(\theta\) is the smaller angle between the vectors when the are drawn with the same initial point.) If either \( \mathbf{a} \) or \( \mathbf{b} \) is \( \mathbf{0} \), we define \(\mathbf{a \cdot b} = 0\).

Two vectors \(\mathbf{a}\) and \(\mathbf{b}\) are orthogonal if and only if \(\mathbf{a \cdot b} = 0\)
The dot product of \(\mathbf{a} = \left \langle a_1, a_2, a_3 \right \rangle\) and \(\mathbf{b} = \left \langle b_1, b_2, b_3 \right \rangle\) is
$$
\mathbf{a \cdot b} = a_1 b_1 + a_2 b_2 + a_3 b_3
$$
If \(\mathbf{a}\),\(\mathbf{b}\), and \(\mathbf{c}\) are vectors in \(V_3\) and \(c\) is a scalar, then

  1. \(\mathbf{a \cdot a}= \left|\mathbf{a}\right|^2\)
  2. \(\mathbf{a \cdot b}= \mathbf{b \cdot a}\)
  3. \(\mathbf{a \cdot} ( \mathbf{b} + \mathbf{c})= \mathbf{a \cdot b} + \mathbf{a \cdot c}\)
  4. \((c\mathbf{a}) \mathbf{\cdot b} = c (\mathbf{a\cdot b}) = \mathbf{a \cdot } (c \mathbf{b})\)
  5. \( \mathbf{0 \cdot a } = 0 \)
The scalar projection of \(\mathbf{b}\) onto \(\mathbf{a}\) is given by:
$$
\text{comp}_{\mathbf{a}}\mathbf{b} = \frac{ \mathbf{a \cdot b}}{\left| \mathbf{a} \right|}
$$
The vector projection of \(\mathbf{b}\) onto \(\mathbf{a}\) is given by:
$$
\text{proj}_{\mathbf{a}}\mathbf{b} = \left(\frac{ \mathbf{a \cdot b}}{\left| \mathbf{a} \right|}\right) \frac{\mathbf{a}}{\left| \mathbf{a} \right|}= \frac{\mathbf{a \cdot b}}{\left| \mathbf{a} \right|^2}\mathbf{a}
$$

E 13.3 Exercises