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By the distributive law (iii) in the text: \[ (\mathbf{a}+\mathbf{b}) \times(\mathbf{a}-\mathbf{b})=(\mathbf{a}+\mathbf{b}) \times \mathbf{a}-(\mathbf{a}+\mathbf{b}...

Question

By the distributive law (iii) in the text: \[ (\mathbf{a}+\mathbf{b}) \times(\mathbf{a}-\mathbf{b})=(\mathbf{a}+\mathbf{b}) \times \mathbf{a}-(\mathbf{a}+\mathbf{b}) \times \mathbf{b}=\mathbf{a} \times \mathbf{a}+\mathbf{b} \times \mathbf{a}-\mathbf{a} \times \mathbf{b}-\mathbf{b} \times \mathbf{b}=2 \mathbf{b} \times \mathbf{a} \] since $\mathbf{a} \times \mathbf{a}=\mathbf{0}, \mathbf{b} \times \mathbf{b}=\mathbf{0},$ and $-\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{a}$

By the distributive law (iii) in the text: \[ (\mathbf{a}+\mathbf{b}) \times(\mathbf{a}-\mathbf{b})=(\mathbf{a}+\mathbf{b}) \times \mathbf{a}-(\mathbf{a}+\mathbf{b}) \times \mathbf{b}=\mathbf{a} \times \mathbf{a}+\mathbf{b} \times \mathbf{a}-\mathbf{a} \times \mathbf{b}-\mathbf{b} \times \mathbf{b}=2 \mathbf{b} \times \mathbf{a} \] since $\mathbf{a} \times \mathbf{a}=\mathbf{0}, \mathbf{b} \times \mathbf{b}=\mathbf{0},$ and $-\mathbf{a} \times \mathbf{b}=\mathbf{b} \times \mathbf{a}$



Answers

If $ a = \langle 1, 0, 1 \rangle $, $ b = \langle 2, 1, -1 \rangle $, and $ c = \langle 0, 1, 3 \rangle $, show that $ a \times (b \times c) \neq (a \times b) \times c $.


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