Dot product of vectors

  • linear algebra (vectors)

Given two vectors of the same length, their dot product is the result of multiplying the vectors component-wise and summing all the products.

Given vectors \(u \mathrel{\mathop:}=(u_1, u_2, \ldots, u_n)\) and \(v \mathrel{\mathop:}=(v_1, v_2, \ldots, v_n)\), the dot product \(u \cdot v\) is

\[ u_1 \times v_1 + u_2 \times v_2 + \cdots + u_n \times v_n \]

where \(\times\) represents multiplication.

For \(u \mathrel{\mathop:}=(1, 3, 5)\) and \(v \mathrel{\mathop:}=(2, 4, 6)\), we have \(u \cdot v = 1 \times 2 + 3 \times 4 + 5 \times 6 = 44\).

Alternatively, we could have also written this as follows using column vectors:

\[ \begin{pmatrix} 1\\ 3\\ 5\\ \end{pmatrix} \cdot \begin{pmatrix} 2\\ 4\\ 6\\ \end{pmatrix} = 1 \times 2 + 3 \times 4 + 5 \times 6 = 44 \]

Calculate all of the following dot products. If the dot product is not defined, say so.

  1. \((1, 2) \cdot (-1, -2)\)
  2. \((1, 1, 1, 1) \cdot (2, 3, 4, 5)\)
  3. \((1, 2) \cdot (-1, \mathrm{True})\)
  4. \((1, 2) \cdot (7)\)
  5. \((1) \cdot (7)\)