Home » Blog » Compute the cross product of given vectors in terms of the unit coordinate vectors

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

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

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

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):