Question

Given any vector $\overrightarrow{r},|\overrightarrow{r}\times \hat{I}{|}^2+|\overrightarrow{r}\times \hat{j}{|}^2+|\overrightarrow{r}\times \hat{k}{|}^2$ equals.

• A. $4|\overrightarrow{r}{|}^2$
• B. $|\overrightarrow{r}{|}^2$
• C. $2|\overrightarrow{r}{|}^2$
• D. $8|\overrightarrow{r}{|}^2$

Answer 1

The correct option is C $2|\overrightarrow{r}{|}^2$

Let

$\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}$

${\left|\overrightarrow{r}\right|}^2=(\sqrt{1^2+1^2+1^2}{)}^2=3$

$\overrightarrow{r}\times \hat{i}=(\hat{i}+\hat{j}+\hat{k})\times \hat{i}$

$\overrightarrow{r}\times \hat{i}=\hat{i}\times \hat{i}+\hat{j}\times \hat{i}+\hat{k}\times \hat{i}$

but $\overrightarrow{i}\times \overrightarrow{i}=0$

$\overrightarrow{r}\times \hat{i}=-\hat{k}+\hat{j}$

$\overrightarrow{r}\times \hat{i}=\hat{j}-\hat{k}$

$|\overrightarrow{r}\times \hat{i}|=\sqrt{1^2+(-1{)}^2}=\sqrt{2}$

${|\overrightarrow{r}\times \hat{i}|}^2={\sqrt{2}}^2=2$----------(1)

$\overrightarrow{r}\times \hat{j}=(\hat{i}+\hat{j}+\hat{k})\times \hat{j}$

$\overrightarrow{r}\times \hat{j}=\hat{i}\times \hat{j}+\hat{j}\times \hat{j}+\hat{k}\times \hat{j}$

but $\overrightarrow{j}\times \overrightarrow{j}=0$

$\overrightarrow{r}\times \hat{j}=\hat{k}-\hat{i}$

$\overrightarrow{r}\times \hat{j}=-\hat{i}+\hat{k}$

$|\overrightarrow{r}\times \hat{j}|=\sqrt{(-1{)}^2+(1{)}^2}=\sqrt{2}$

${|\overrightarrow{r}\times \hat{j}|}^2={\sqrt{2}}^2=2$----------(2)

$\overrightarrow{r}\times \hat{k}=(\hat{i}+\hat{j}+\hat{k})\times \hat{k}$

$\overrightarrow{r}\times \hat{k}=\hat{i}\times \hat{k}+\hat{j}\times \hat{k}+\hat{k}\times \hat{k}$

but $\overrightarrow{k}\times \overrightarrow{k}=0$

$\overrightarrow{r}\times \hat{k}=-\hat{j}+\hat{i}$

$\overrightarrow{r}\times \hat{k}=\hat{i}-\hat{j}$

$|\overrightarrow{r}\times \hat{k}|=\sqrt{(1{)}^2+(1{)}^2}=\sqrt{2}$

${|\overrightarrow{r}\times \hat{k}|}^2={\sqrt{2}}^2=2$----------(3)

Adding eq (1) (2) and (3)

${|\overrightarrow{r}\times \hat{i}|}^2+{|\overrightarrow{r}\times \hat{j}|}^2+{|\overrightarrow{r}\times \hat{k}|}^2=2+2+2$

${|\overrightarrow{r}\times \hat{i}|}^2+{|\overrightarrow{r}\times \hat{j}|}^2+{|\overrightarrow{r}\times \hat{k}|}^2=6$

${|\overrightarrow{r}\times \hat{i}|}^2+{|\overrightarrow{r}\times \hat{j}|}^2+{|\overrightarrow{r}\times \hat{k}|}^2=2\times 3$

${|\overrightarrow{r}\times \hat{i}|}^2+{|\overrightarrow{r}\times \hat{j}|}^2+{|\overrightarrow{r}\times \hat{k}|}^2=2{\left|\overrightarrow{r}\right|}^2$

Hence option C is correct