Three Dimensional Geometry Question 103
Question: If $ l_1,,m_1,,n_1 $ and $ l_2,m_2,n_2 $ are the direction cosines of two perpendicular lines, then the direction cosine of the line which is perpendicular to both the lines, will be
Options:
A) $ (m_1n_2-m_2n_1),(n_1l_2-n_2l_1),,(l_1m_2-l_2m_1) $
B) $ (l_1l_2-m_1m_2),,(m_1m_2-n_1n_2),,(n_1n_2-l_1l_2) $
C) $ \frac{1}{\sqrt{l_1^{2}+m_1^{2}+n_1^{2}}},\frac{1}{\sqrt{l_2^{2}+m_2^{2}+n_2^{2}}},\frac{1}{\sqrt{3}} $
D) $ \frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}},\frac{1}{\sqrt{3}} $
Show Answer
Answer:
Correct Answer: A
Solution:
Let lines are $ l_1x+m_1y+n_1z+d=0 $ ?..(i) and $ l_2x+m_2y+n_2z+d=0 $ …..(ii) If $ lx+my+nz+d=0 $ is perpendicular to (i) and (ii), then, $ ll_1+mm_1+nn_1=0,ll_2+mm_2+nn_2=0, $
$ \Rightarrow \text{ },\frac{l}{m_1n_2-m_2n_1}=\frac{m}{n_1l_2-l_1n_2}=\frac{n}{l_1m_2-l_2m_1}=d $ Therefore, direction cosines are $ (m_1n_2-m_2n_1),(n_1l_2-l_1n_2),,(l_1m_2-l_2m_1) $ .
BETA
