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

by RoRi

Let A,B,C,D \in \mathbb{R}^3 such that

    \[ A \times C \cdot B = 5, \quad A \times D \cdot B = 3, \quad C + D = \mathbf{i} + 2 \mathbf{j} + \mathbf{k}, \quad C - D = \mathbf{i} - \mathbf{k}. \]

In terms of the unit coordinate vectors \mathbf{i}, \mathbf{j}, \mathbf{k} compute the cross product

    \[ (A\times B) \times (C \times D). \]

Using part (a) of the previous exercise and equation (13.10) on page 490 of Apostol (A \times B \cdot C = A \cdot B \times C) we compute

    \begin{align*} (A \times B) \times (C \times D) &= (A \times B \cdot D)C - (A \times B \cdot C)D \\ &= (A \cdot B \times C) C - (A \cdot B \times C) D \\ &= (A \cdot C \times B) D - (A \cdot B \times B) C \\ &= 5D - 3C. \end{align*}

Then we use the other given relations

    \begin{align*} && C + D &= \mathbf{i} + 2 \mathbf{j} + \mathbf{k} \\ \implies && 2D &= 2 \mathbf{j} + 2 \mathbf{k} \\ \implies && 5D &= 5\mathbf{j} + 5 \mathbf{k} \\ && C-D &= \mathbf{i} - \mathbf{k} \\ \implies && 2C &= 2 \mathbf{i} + 2 \mathbf{j} \\ \implies && 3C &= 3 \mathbf{i} + 3 \mathbf{j}. \end{align*}

Which all implies

    \[ 5D - 3C = -3 \mathbf{i} + 2 \mathbf{j} + 5 \mathbf{k}. \]

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