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Compute some dot products of vectors in C2

by RoRi

Let

    \[ A = (1,i), \qquad B = (i,-i), \qquad C = (2i,1) \]

be vectors in \mathbb{C}^2. Compute the following dot products.

  1. A \cdot B;
  2. B \cdot A;
  3. (iA) \cdot B;
  4. A \cdot (iB);
  5. (iA) \cdot (iB);
  6. B \cdot C;
  7. A \cdot C;
  8. (B+C) \cdot A;
  9. (A-C) \cdot B;
  10. (A - iB) \cdot (A + iB).
  1. We compute,

        \[ A \cdot B = (1)(-i) + (i)(i) = -1 -i. \]

  2. We compute,

        \[ B \cdot A = (i)(1) + (-i)(-i) = -1 + i. \]

  3. We compute,

        \[ (iA) \cdot B = (i, -1) \cdot (i, -i) = (i)(-i) + (-1)(i) = 1 - i. \]

  4. We compute,

        \[ A \cdot (iB) = (1,i) \cdot (-1,1) = (1)(-1) + (i)(1) = -1 + i. \]

  5. We compute,

        \[ (iA) \cdot (iB) = (i,-1)\cdot (-1,1) = (i)(-1) + (-1)(1) = -1-i. \]

  6. We compute,

        \[ B \cdot C = (i)(-2i) + (-i)(1) = 2 - i. \]

  7. We compute,

        \[ A \cdot C = (1)(-2i) + (i)(1) = -i. \]

  8. We compute,

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

  9. We compute,

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

  10. We compute,

        \[ (A-iB)\cdot(A+iB) = A\cdot A + A \cdot i B - iB \cdot A - B\cdot B = 2 + (-1+i) - (-1-i) -2 = 2i. \]

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