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Compute some dot products of given vectors

Consider the three vectors

    \[ A = (1,2,3,4), \qquad B = (-1,2,-3,0), \qquad C = (0,1,0,1) \]

in \mathbb{R}^4. Compute:

  1. A \cdot B;
  2. B \cdot C;
  3. A \cdot C;
  4. A \cdot (B+C);
  5. (A-B)\cdot C.

  1. We compute

        \[ A \cdot B = (1,2,3,4) \cdot (-1,2,-3,0) = (1)(-1) + (2)(2) + (3)(-3) + (4)(0) = -1 +4 - 9 + 0 = -6. \]

  2. We compute

        \[ B \cdot C = (-1,2,-3,0) \cdot (0,1,0,1) = (-1)(0) + (2)(1) + (-3)(0)+ (0)(1) = 0 + 2 + 0 + 0 = 2. \]

  3. We compute

        \[ A \cdot C = (1,2,3,4) \cdot (0,1,0,1) = (1)(0) + (2)(1) + (3)(0) + (4)(1) = 0 + 2 + 0 + 4 = 6. \]

  4. We compute

        \begin{align*}  A \cdot (B+C) &= (1,2,3,4) \cdot ((-1,2,-3,0) + (0,1,0,1)) \\  &= (1,2,3,4) \cdot (-1,3, -3, 1) \\  &= (1)(-1) + (2)(3) + (3)(-3) + (4)(1) \\  &= -1 + 6 - 9 + 4 \\  &= 0. \end{align*}

  5. We compute

        \begin{align*}  (A - B) \cdot C &= ((1,2,3,4) - (-1,2,-3,0)) \cdot (0,1,0,1) \\  &= (2,0,6,4) \cdot (0,1,0,1) \\  &= (2)(0) + (0)(1) + (6)(0) + (4)(1) \\  &= 0 + 0 + 0 + 4\\  &= 4. \end{align*}

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