3D Geometry 3 Question 58
####58. The unit vector perpendicular to both $L_{1}$ and $L_{2}$ is
(a) $\frac{-\hat{\mathbf{i}}+7 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}}{\sqrt{99}}$
(b) $\frac{-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}}{5 \sqrt{3}}$
(c) $\frac{-\hat{\mathbf{i}}+7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}}{5 \sqrt{3}}$
(d) $\frac{7 \hat{\mathbf{i}}-7 \hat{\mathbf{j}}-\hat{\mathbf{k}}}{\sqrt{99}}$
Assertion and Reason
For the following questions, choose the correct answer from the codes (a), (b), (c) and (d) defined as follows.
(a) Statement I is true, Statement II is also true; Statement II is the correct explanation of Statement I
(b) Statement I is true, Statement II is also true; Statement II is not the correct explanation of Statement I
(c) Statement I is true; Statement II is false
(d) Statement I is false; Statement II is true
Show Answer
Answer:
Correct Answer: 58. (b)
Solution:
- The equations of given lines in vector form may be written as $\quad L_{1}: \overrightarrow{\mathbf{r}}=(\hat{\mathbf{i}}-2 \hat{\mathbf{j}}-\hat{\mathbf{k}})+\lambda(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})$
and $\quad L_{2}: \overrightarrow{\mathbf{r}}=(2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})+\mu(\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$
Since, the vector is perpendicular to both $L_{1}$ and $L_{2}$.
$ \left|\begin{array}{ccc} \hat{\mathbf{i}} & \hat{\mathbf{j}} & \hat{\mathbf{k}} \\ 3 & 1 & 2 \\ 1 & 2 & 3 \end{array}\right|=-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}} $
$\therefore$ Required unit vector
$ =\frac{(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}})}{\sqrt{(-1)^{2}+(-7)^{2}+(5)^{2}}}=\frac{1}{5 \sqrt{3}}(-\hat{\mathbf{i}}-7 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}) $
