Write the value of [ i - j j - k k - i ]. - Vidyadip

Question

Write the value of $\big[\hat{\text{i}}-\hat{\text{j }}\hat{\text{j}}-\hat{\text{k }}\hat{\text{k}}-\hat{\text{i}}\big].$

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Answer

We have
$\big[\hat{\text{i}}-\hat{\text{j }}\hat{\text{j}}-\hat{\text{k }}\hat{\text{k}}-\hat{\text{i}}\big]=\big[\big(\hat{\text{i}}-\hat{\text{j}}\big)\times\big(\hat{\text{j}}-\hat{\text{k}}\big)\big].\big(\hat{\text{k}}-\hat{\text{i}}\big)$
$\big(\therefore\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]=\big(\vec{\text{a}}\times\vec{\text{b}}\big).\vec{\text{c}}\big)$
$\big[\big(\hat{\text{i}}\times\hat{\text{j}}\big)-\big(\hat{\text{i}}\times\hat{\text{k}}\big)-\big(\hat{\text{j}}\times{\hat{\text{j}}}\big)+\big(\hat{\text{j}}\times\hat{\text{k}}\big)\big].\big(\hat{\text{k}}-\hat{\text{i}}\big)$
$=\big[\hat{\text{k}}+\hat{\text{j}}+\hat{\text{i}}\big].\big(\hat{\text{k}}-\hat{\text{i}}\big)$
$=\big[\big(\hat{\text{k}}.\hat{\text{k}}\big)-\big(\hat{\text{k}}.\hat{\text{i}}\big)+\big(\hat{\text{j}}.\hat{\text{k}}\big)-\big(\hat{\text{j}}.\hat{\text{i}}\big)+\big(\hat{\text{i}}.\hat{\text{k}}\big)-\big(\hat{\text{i}}.\hat{\text{i}}\big)\big]$
$=1-0+0-0+0-1=0$

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