Angle between Two Line Segments

Q. If $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2,)$ are d.c.'s of $\overline{OA,}$ $\overline{OB}$ such that $\angle AOB=\theta$ where ‘O’ is the origin, then the d.c.’s of the internal bisector of the angle $\angle AOB$ are

1. $\frac{l_1+l_2}{2sin\frac{\theta }{2}},\frac{m_1+m_2}{2sin\frac{\theta }{2}},\frac{n_1+n_2}{2sin\frac{\theta }{2}}$
2. $\frac{l_1+l_2}{2cos\frac{\theta }{2}},\frac{m_1+m_2}{2cos\frac{\theta }{2}},\frac{n_1+n_2}{2cos\frac{\theta }{2}}$
3. $\frac{l_1-l_2}{2sin\frac{\theta }{2}},\frac{m_1-m_2}{2sin\frac{\theta }{2}},\frac{n_1-n_2}{2sin\frac{\theta }{2}}$
4. $\frac{l_1-l_2}{2cos\frac{\theta }{2}},\frac{m_1-m_2}{2cos\frac{\theta }{2}},\frac{n_1-n_2}{2cos\frac{\theta }{2}}$

Q. The lines $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{-1},\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{4}$ are

1. parallel lines
2. intersecting lines
3. perpendicular lines
4. skew lines

Q. If the angle between two intersecting lines having direction ratios $(5,7,3)$ & $(3,4,5)$ respectively can be given by
${\cos }^{-1}\left(\frac{58}{\sqrt{b}}\right)$,
then what will be the value of $b$ ?

1. 4052
2. 3971
3. 4167
4. 4150

Q.

The angle between any two diagonals of a cube is:

1. $co{s}^{-1}\left(\frac{1}{3}\right)$

2. $co{s}^{-1}\left(\frac{1}{4}\right)$

3. $co{s}^{-1}\left(\frac{1}{2}\right)$

4. $\frac{\pi }{2}$

Q. If the direction cosines of a variable line in two adjacent positions be l, m, n and l + a, m + b, n + c and the small angle between the two positions be $\theta$, then :

1. ${\theta }^2=a^2+b^2+c^2$
2. $|\theta |=|a|+|b|+|c|$
3. ${\theta }^3=a^3+b^3+c^3$
4. $\theta =a+b+c$

Q. The value of "b" is equal to , if the angle between two lines having direction ratios 5, 7, 3 & 3, 4, 5 respectively can be given by $co{s}^{-1}\left(\frac{58}{\sqrt{b}}\right)$

1. 4150
2. 4052
3. 3971
4. 4167

Q.

Find the angle of intersection of the curves $y=4-x^{2}andy=x^2$

Q. Let two lines $L_1:\frac{x-3}{3}=\frac{y-2}{-4}=\frac{z}{-1}$ and $L_2:\frac{x-3}{4}=\frac{y-2}{1}=\frac{z}{-3}$ and $\frac{x-3}{a}=\frac{y-2}{6}=\frac{z}{b},(a,b\in R)$ is the acute angle biscetor between lines $L_1$ and $L_2.$ Then the value of $|a-b|=$

Q. If $(l_1,m_1,n_1)$ and $(l_2,m_2,n_2,)$ are d.c.'s of $\overline{OA,}$ $\overline{OB}$ such that $\angle AOB=\theta$ where ‘O’ is the origin, then the d.c.’s of the internal bisector of the angle $\angle AOB$ are

1. $\frac{l_1+l_2}{2cos\frac{\theta }{2}},\frac{m_1+m_2}{2cos\frac{\theta }{2}},\frac{n_1+n_2}{2cos\frac{\theta }{2}}$
2. $\frac{l_1-l_2}{2sin\frac{\theta }{2}},\frac{m_1-m_2}{2sin\frac{\theta }{2}},\frac{n_1-n_2}{2sin\frac{\theta }{2}}$
3. $\frac{l_1+l_2}{2sin\frac{\theta }{2}},\frac{m_1+m_2}{2sin\frac{\theta }{2}},\frac{n_1+n_2}{2sin\frac{\theta }{2}}$
4. $\frac{l_1-l_2}{2cos\frac{\theta }{2}},\frac{m_1-m_2}{2cos\frac{\theta }{2}},\frac{n_1-n_2}{2cos\frac{\theta }{2}}$

Q. The lines $\frac{x+1}{1}=\frac{y-1}{2}=\frac{z-2}{-1},\frac{x-1}{2}=\frac{y}{1}=\frac{z+1}{4}$ are

1. skew lines
2. parallel lines
3. perpendicular lines
4. intersecting lines