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Question

# Let $\mathbf{u}=(3,2,-1), \mathbf{v}=(0,2,-3)$, and $\mathbf{w}=(2,6,7)$. Compute the indicated vectors. (a) $(\mathbf{u} \times \mathbf{v}) \times(\mathbf{v} \times \mathbf{w})$ (b) $\mathbf{u} \times(\mathbf{v}-2 \mathbf{w})$ (c) $(\mathbf{u} \times \mathbf{v})-2 \mathbf{w}$.

Solution

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Step 1
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(a)

\begin{align*} \textbf{u} \times \textbf{v} &= -4\textbf{i} + 9\textbf{j} + 6\textbf{k} \\ \textbf{v} \times \textbf{w} &= 32\textbf{i} - 6\textbf{j} - 4\textbf{k} \\ \\ (\textbf{u} \times \textbf{v}) \times (\textbf{v} \times \textbf{w}) &= \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ -4 & 9 & 6 \\ 32 & -6 & -4 \end{vmatrix} \\ &= (-36+36)\textbf{i} - (16-192)\textbf{j} + (24-288)\textbf{k} \\ &= 176\textbf{j} - 264\textbf{k} \end{align*}

(b)

\begin{align*} \textbf{v} - 2\textbf{w} &= (0,2,-3) - (4,12,14) \\ &= (-4,-10,-17) \\ \\ \textbf{u} \times (\textbf{v} - 2\textbf{w}) &= \begin{vmatrix} \textbf{i} & \textbf{j} & \textbf{k} \\ 3 & 2 & -1 \\ -4 & -10 & -17 \end{vmatrix} \\ &= (-34-10)\textbf{i} - (-51-4)\textbf{j} + (-30+8)\textbf{k} \\ &= -44\textbf{i} + 55\textbf{j} - 22\textbf{k} \end{align*}

(c)

\begin{align*} \textbf{u} \times \textbf{v} &= -4\textbf{i} + 9\textbf{j} + 6\textbf{k} \\ (\textbf{u} \times \textbf{v}) - 2\textbf{w} &= (-4\textbf{i} + 9\textbf{j} + 6\textbf{k}) - (4\textbf{i} + 12\textbf{j} + 14\textbf{k}) \\ &= -8\textbf{i} - 3\textbf{j} - 8\textbf{k} \end{align*}

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