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Question Number 225 by 123456 last updated on 25/Jan/15

$$\mathbf{u}=2\mathbf{i}+3\mathbf{j}+5\mathbf{k}$$$$\mathbf{v}=3\mathbf{i}+2\mathbf{j}-5\mathbf{k}$$$$\mathbf{w}=-2\mathbf{i}+3\mathbf{j}+\mathbf{k}$$$$\mathrm{find}$$$$\mathbf{u}\cdot \left(\mathbf{v}\times \mathbf{w}\right)+\left(\mathbf{u}\times \mathbf{v}\right)\cdot \left(\mathbf{u}\times \mathbf{w}\right)$$

Answered by mreddy last updated on 16/Dec/14

$$\mathbf{v}\times \mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 3& 2& -5\\ -2& 3& 1\end{array}\right|=17\mathbf{i}+7\mathbf{j}+13k$$$$\mathbf{u}\times \mathbf{w}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 3& 5\\ -2& 3& 1\end{array}\right|=-12\mathbf{i}-12\mathbf{j}+12k$$$$\mathbf{u}\times \mathbf{v}=\left|\begin{array}{ccc}\mathbf{i}& \mathbf{j}& \mathbf{k}\\ 2& 3& 5\\ 3& 2& -5\end{array}\right|=-25\mathbf{i}+25\mathbf{j}-5k$$$$\left(\mathbf{u}\times \mathbf{v}\right)\cdot \left(\mathbf{u}\times \mathbf{w}\right)=25\times 12-25\times 12-60=-60$$$$\mathbf{u}\cdot \left(\mathbf{v}\times \mathbf{w}\right)=34+21+65=120$$$$\mathbf{u}\cdot \left(\mathbf{v}\times \mathbf{w}\right)+\left(\mathbf{u}\times \mathbf{v}\right)\cdot \left(\mathbf{u}\times \mathbf{w}\right)=120-60=60$$

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