Q129 / 160 · AP EAMCET 2022 (06 Jul Shift 1) · Physics
Two position vectors are given by $\overrightarrow{r_1}=(1,1,1)$ and $\overrightarrow{r_2}=(1,-1,1)$. The unit vector in the direction of $\overrightarrow{r_1} \times \overrightarrow{r_2}$
- Correct answerA$\frac{\hat{\mathrm{i}}}{\sqrt{2}}-\frac{\hat{\mathrm{k}}}{\sqrt{2}}$
- B$-\frac{\hat{\mathrm{i}}}{\sqrt{2}}-\frac{\hat{\mathrm{k}}}{\sqrt{2}}$
- C$\frac{\hat{\mathrm{i}}}{\sqrt{2}}+\frac{\hat{\mathrm{k}}}{\sqrt{2}}$
- D$-\frac{\hat{\mathrm{i}}}{\sqrt{2}}+\frac{\hat{\mathrm{k}}}{\sqrt{2}}$
Solution
We have
$\begin{aligned}
& \overline{\mathrm{r}_1} \times \overline{\mathrm{r}_2}=\left|\begin{array}{ccc}
\hat{\mathrm{i}} & \hat{\mathrm{j}} & \hat{\mathrm{k}} \\
1 & 1 & 1 \\
1 & -1 & 1
\end{array}\right| \\
& =\hat{\mathrm{i}}(2)+\hat{\mathrm{j}}(0)+\hat{\mathrm{k}}(-2)=2 \hat{\mathrm{i}}-2 \hat{\mathrm{k}}
\end{aligned}$
So, unit vector in the direction of
$\left(\overrightarrow{\mathrm{r}_1} \times \overrightarrow{\mathrm{r}_2}\right)=\frac{2 \hat{\mathrm{i}}-2 \hat{\mathrm{k}}}{\sqrt{2^2+(-2)^2}}=\frac{\hat{\mathrm{i}}}{\sqrt{2}}-\frac{\hat{\mathrm{k}}}{\sqrt{2}}$
Hence, the correct answer is option: 'A'.