AP EAMCET · Maths · Three Dimensional Geometry
If \(l_1, m_1, n_1\) and \(l_2, m_2, n_2\) are direction cosines of \(\mathbf{O A}\) and \(O B\) such that \(\angle A O B=\theta\), where \(O\) is the origin, then the direction cosines of the internal angular bisector of \(\angle A O B\) are
- A \(\frac{I_1+I_2}{2 \sin \frac{\theta}{2}}, \frac{m_1+m_2}{2 \sin \frac{\theta}{2}}, \frac{n_1+n_2}{2 \sin \frac{\theta}{2}}\)
- B \(\frac{l_1-l_2}{2 \cos \frac{\theta}{2}}, \frac{m_1-m_2}{2 \cos \frac{\theta}{2}}, \frac{n_1-n_2}{2 \cos \frac{\theta}{2}}\)
- C \(\frac{l_1-l_2}{2 \sin \frac{\theta}{2}}, \frac{m_1-m_2}{2 \sin \frac{\theta}{2}}, \frac{n_1-n_2}{2 \sin \frac{\theta}{2}}\)
- D \(\frac{l_1+I_2}{2 \cos \frac{\theta}{2}}, \frac{m_1+m_2}{2 \cos \frac{\theta}{2}}, \frac{n_1+n_2}{2 \cos \frac{\theta}{2}}\)
Answer & Solution
Correct Answer
(D) \(\frac{l_1+I_2}{2 \cos \frac{\theta}{2}}, \frac{m_1+m_2}{2 \cos \frac{\theta}{2}}, \frac{n_1+n_2}{2 \cos \frac{\theta}{2}}\)
Step-by-step Solution
Detailed explanation
\(\because l_1 l_2+m_1 m_2+n_1 n_2=\cos \theta\) Through origin \(O\) draw two lines parallel to given lines and take two points on each at a distance \(r\) from \(O\) and a point \(R\) on \(Q O\) produced so that \(O R=r\) Then, the coordinates of \(P, Q\) and \(R\) are…
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