Which of the following relations is correct?

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(A) In vector algebra,for unit vectors $\hat{i}, \hat{j}, \hat{k}$ along the $x, y, z$ axes respectively:$1$. Cross product properties: $\hat{i} 	ime

Vedclass · Accepted Answer

$\hat{j} 	imes \hat{k} = \hat{i}$

Vedclass · Answer

(A) In vector algebra,for unit vectors $\hat{i}, \hat{j}, \hat{k}$ along the $x, y, z$ axes respectively:$1$. Cross product properties: $\hat{i} 	imes \hat{j} = \hat{k}$,$\hat{j} 	imes \hat{k} = \hat{i}$,and $\hat{k} 	imes \hat{i} = \hat{j}$. Also,$\hat{i} 	imes \hat{i} = \hat{j} 	imes \hat{j} = \hat{k} 	imes \hat{k} = 0$.$2$. Dot product properties: $\hat{i} \cdot \hat{i} = \hat{j} \cdot \hat{j} = \hat{k} \cdot \hat{k} = 1$ and $\hat{i} \cdot \hat{j} = \hat{j} \cdot \hat{k} = \hat{k} \cdot \hat{i} = 0$.Comparing these with the options:Option $A$: $\hat{j} 	imes \hat{k} = \hat{i}$ is correct.Option $B$: $\hat{i} \cdot \hat{i} = 1$ (given $0$,so incorrect).Option $C$: $\hat{j} 	imes \hat{j} = 0$ (given $1$,so incorrect).Option $D$: $\hat{k} \cdot \hat{i} = 0$ (given $1$,so incorrect).Therefore,the correct relation is $\hat{j} 	imes \hat{k} = \hat{i}$.

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