Question

If $\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}$, then $(\overrightarrow{a}\times \hat{i}).(\hat{i}+\hat{j})+(\overrightarrow{a}\times \hat{j}).(\hat{j}+\hat{k})+(\overrightarrow{a}\times \hat{k}).(\hat{k}\times \hat{i})=$

• A. $x-y+z$
• B. $x+y+z$
• C. $x+y-z$
• D. $-x+y+z$

Answer 1

The correct option is B $x+y+z$

Given, $\overrightarrow{a}=x\hat{i}+y\hat{j}+z\hat{k}$

$\overrightarrow{a}\times \hat{i}=-y\hat{k}+z\hat{j}$

$(\overrightarrow{a}\times \overrightarrow{i}).(\overrightarrow{i}\times \overrightarrow{j})=z$
$(\overrightarrow{a}\times \overrightarrow{j}).\overrightarrow{k}=x$
$(\overrightarrow{a}\times \overrightarrow{k}).(\overrightarrow{k}\times \overrightarrow{i})=y$
Thus the value of $(\overrightarrow{a}\times \hat{i}).(\hat{i}+\hat{j})+(\overrightarrow{a}\times \hat{j}).(\overrightarrow{j}+\hat{k})+(\hat{a}\times \overrightarrow{k}).(\overrightarrow{k}\times \overrightarrow{i})$ is $x+y+z$.