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Express the cross product of given vectors in terms of the unit coordinate vectors

Let

    \[ A = 2 \mathbf{i} + 5 \mathbf{j} + 3 \mathbf{k}, \quad B = 2 \mathbf{i} + 7 \mathbf{j} + 4 \mathbf{k}, \quad C = 3 \mathbf{i} + 3 \mathbf{j} + 6 \mathbf{k}. \]

Find (A - C) \times (B - A) in terms of \mathbf{i}, \mathbf{j}, \mathbf{k}.


First, we have

    \begin{align*}  A - C &= - \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \\  B - A &= 2\mathbf{j} + \mathbf{k}. \end{align*}

So, the cross product is given by

    \begin{align*}  (A - C) \times (B - A) &= \begin{vmatrix*} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ -1 & 2 & -3 \\ 0 & 2 & 1 \end{vmatrix*} \\[9pt]  &= \begin{vmatrix*} 2 & -3 \\ 2 & 1 \end{vmatrix*} \mathbf{i} - \begin{vmatrix*} -1 & -3 \\ 0 & 1 \end{vmatrix*} \mathbf{j} + \begin{vmatrix*} -1 & 2 \\ 0 & 2 \end{vmatrix*} \mathbf{k} \\[9pt]  &= 8 \mathbf{i} + \mathbf{j} - 2 \mathbf{k} \end{align*}

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