(i+j) (j+k) (k+i) is equal to - Vidyadip

Question

$(\hat{i}+\hat{j}) \times(\hat{j}+\hat{k}) \cdot(\hat{k}+\hat{i})$ is equal to

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Answer

$\text { (c) : }(\hat{i}+\hat{j}) \times(\hat{j}+\hat{k}) \cdot(\hat{k}+\hat{i})=(\hat{i} \times \hat{j}+\hat{i} \times \hat{k}+\hat{j} \times \hat{k}) \cdot(\hat{k}+\hat{i})$
$=(\hat{k}-\hat{j}+\hat{i}) \cdot(\hat{k}+\hat{i})=\hat{k} \cdot \hat{k}+\hat{i} \cdot \hat{i} \quad(\because \hat{i} \cdot \hat{j}=\hat{j} \cdot \hat{k}=\hat{k} \cdot \hat{i}=0)$
$=|\hat{k}|^2+|\hat{i}|^2=1+1=2$

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