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Compute the cross products of given vectors

Consider the vectors

    \[ A = -\mathbf{i} + 2 \mathbf{k}, \qquad B = 2 \mathbf{i} + \mathbf{j} - \mathbf{k}, \qquad C = \mathbf{i} + 2 \mathbf{j} + 2 \mathbf{k}. \]

Compute each of the following.

  1. A \times B;
  2. B \times C;
  3. C \times A;
  4. A \times (C \times A);
  5. (A \times B) \times C;
  6. A \times (B \times C);
  7. (A \times C) \times B;
  8. (A + B) \times (A - C);
  9. (A \times B) \times (A \times C).

  1. We compute

        \begin{align*}  A \times B &= \left( \begin{vmatrix*}[r] a_2 & a_3 \\ b_2 & b_3 \end{vmatrix*}, \begin{vmatrix*}[r] a_3 & a_1 \\ b_3 & b_1 \end{vmatrix*}, \begin{vmatrix*}[r] a_1 & a_2 \\ b_1 & b_2 \end{vmatrix*} \right) \\[9pt]  &= \left( \begin{vmatrix*}[r] 0 & 2 \\ 1 & -1 \end{vmatrix*}, \begin{vmatrix*}[r] 2 & -1 \\ -1 & 2 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & 0 \\ 2 & 1 \end{vmatrix*} \right) \\[9pt]  &= (0-2, 4 + (-1), -1-0) \\[9pt]  &= (-2,3,-1). \end{align*}

  2. We compute

        \begin{align*}  B \times C &= \left( \begin{vmatrix*}[r] b_2 & b_3 \\ c_2 & c_3 \end{vmatrix*}, \begin{vmatrix*}[r] b_3 & b_1 \\ c_3 & c_1 \end{vmatrix*}, \begin{vmatrix*}[r] b_1 & b_2 \\ c_1 & c_2 \end{vmatrix*} \right) \\[9pt]  &= \left( \begin{vmatrix*}[r] 1 & -1 \\ 2 & 2 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & 2 \\ 2 & 1 \end{vmatrix*}, \begin{vmatrix*}[r] 2 & 1 \\ 1 & 2 \end{vmatrix*} \right) \\[9pt]  &= (2+2,-1-4,4-1) \\[9pt]  &= (4,-5,3). \end{align*}

  3. We compute

        \begin{align*}  C \times A &= \left( \begin{vmatrix*}[r] c_2 & c_3 \\ a_2 & a_3 \end{vmatrix*}, \begin{vmatrix*}[r] c_3 & c_1 \\ a_3 & a_1 \end{vmatrix*}, \begin{vmatrix*}[r] c_1 & c_2 \\ a_1 & a_2 \end{vmatrix*} \right) \\[9pt]  &= \left( \begin{vmatrix*}[r] 2 & 2 \\ 0 & 2 \end{vmatrix*}, \begin{vmatrix*}[r] 2 & 1 \\ 2 & -1 \end{vmatrix*}, \begin{vmatrix*}[r] 2 & 2 \\ -1 & 0 \end{vmatrix*} \right) \\[9pt]  &= (4,-2-2,0+2) \\[9pt]  &= (4,-4,2). \end{align*}

  4. Using part (c) we compute

        \begin{align*}  A \times (C \times A) &= A \times (4,-4,2) \\[9pt]  &= \left( \begin{vmatrix*} 0 & 2 \\ -4 & 2 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & -2 \\ 2 &1 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & 0 \\ 4 & -4 \end{vmatrix*} \right) \\[9pt]  &= (8,10,4). \end{align*}

  5. Using part (a) we compute

        \begin{align*}  (A \times B) \times C &= (-2,3,-1) \times C \\[9pt]  &= \left( \begin{vmatrix*} 3 & -1 \\ 2 & 2 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & -2 \\ 2 &1 \end{vmatrix*}, \begin{vmatrix*}[r] -2 & 3 \\ 1 & 2 \end{vmatrix*} \right) \\[9pt]  &= (8,3,-7). \end{align*}

  6. Using part (b) we compute

        \begin{align*}  A \times (B \times C) &= A \times (4,-5,3) \\[9pt]  &= \left( \begin{vmatrix*} 0 & 2 \\ -5 & 3 \end{vmatrix*}, \begin{vmatrix*}[r] 2 & -1 \\ 3 & 4 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & 0 \\ 4 & -5 \end{vmatrix*} \right) \\[9pt]  &= (10,11,5). \end{align*}

  7. Using part (c) we compute

        \begin{align*}  (A \times C) \times B &= (-4,4,-2) \times B \\[9pt]  &= \left( \begin{vmatrix*} 4 & -2 \\ 1 & -1 \end{vmatrix*}, \begin{vmatrix*}[r] -2 & -4 \\ -1 & 2 \end{vmatrix*}, \begin{vmatrix*}[r] -4 & 4 \\ 2 & 1 \end{vmatrix*} \right) \\[9pt]  &= (-2,-8,-12). \end{align*}

  8. We compute

        \begin{align*}  (A + B) \times (A - C) &= (1,1,1) \times (-2,-2,0) \\[9pt]  &= \left( \begin{vmatrix*} 1 & 1 \\ -2 & 0 \end{vmatrix*}, \begin{vmatrix*}[r] 1 & 1 \\ 0 & -2 \end{vmatrix*}, \begin{vmatrix*}[r] 1 & 1 \\ -2 & -2 \end{vmatrix*} \right) \\[9pt]  &= (2,-2,0). \end{align*}

  9. We compute

        \begin{align*}  (A \times B) \times (A \times C) &= (-2,3,-1) \times (-4,4,-2) \\[9pt]  &= \left( \begin{vmatrix*} 3 & -1 \\ 4 & -2 \end{vmatrix*}, \begin{vmatrix*}[r] -1 & -2 \\ -2 & -4 \end{vmatrix*}, \begin{vmatrix*}[r] -2 & 3 \\ -4 & 4 \end{vmatrix*} \right) \\[9pt]  &= (-2,0,4). \end{align*}

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