Question

# Write the value of $\left[\stackrel{^}{i}-\stackrel{^}{j}\stackrel{^}{j}-\stackrel{^}{k}\stackrel{^}{k}-\stackrel{^}{i}\right].$

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Solution

## $\mathrm{We}\mathrm{have}\phantom{\rule{0ex}{0ex}}\left[\stackrel{^}{i}-\stackrel{^}{j}\stackrel{^}{j}-\stackrel{^}{k}\stackrel{^}{k}-\stackrel{^}{i}\right]=\left[\left(\stackrel{^}{i}-\stackrel{^}{j}\right)×\left(\stackrel{^}{j}-\stackrel{^}{k}\right)\right]·\left(\stackrel{^}{k}-\stackrel{^}{i}\right)\left(\because \left[\stackrel{\to }{a}\stackrel{\to }{b}\stackrel{\to }{c}\right]=\left(\stackrel{\to }{a}×\stackrel{\to }{b}\right).\stackrel{\to }{c}\right)\phantom{\rule{0ex}{0ex}}=\left[\left(\stackrel{^}{i}×\stackrel{^}{j}\right)-\left(\stackrel{^}{i}×\stackrel{^}{k}\right)-\left(\stackrel{^}{j}×\stackrel{^}{j}\right)+\left(\stackrel{^}{j}×\stackrel{^}{k}\right)\right]·\left(\stackrel{^}{k}-\stackrel{^}{i}\right)\phantom{\rule{0ex}{0ex}}=\left[\stackrel{^}{k}+\stackrel{^}{j}+\stackrel{^}{i}\right]·\left(\stackrel{^}{k}-\stackrel{^}{i}\right)\phantom{\rule{0ex}{0ex}}=\left[\left(\stackrel{^}{k}·\stackrel{^}{k}\right)-\left(\stackrel{^}{k}·\stackrel{^}{i}\right)+\left(\stackrel{^}{j}·\stackrel{^}{k}\right)-\left(\stackrel{^}{j}·\stackrel{^}{i}\right)+\left(\stackrel{^}{i}·\stackrel{^}{k}\right)-\left(\stackrel{^}{i}·\stackrel{^}{i}\right)\right]\phantom{\rule{0ex}{0ex}}=1-0+0-0+0-1=0\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$

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