Angle between Two Line Segments

Q. The line $\frac{x+6}{5}=\frac{y+10}{3}=\frac{z+14}{8}$ is the hypotenuse of an isosceles right-angled triangle whose opposite vertex is $(7,2,4)$. Then which of the following is not the side of the triangle?

1. $\frac{x-7}{2}=\frac{y-2}{-3}=\frac{z-4}{6}$
2. $\frac{x-7}{3}=\frac{y-2}{6}=\frac{z-4}{2}$
3. $\frac{x-7}{3}=\frac{y-2}{5}=\frac{z-4}{-1}$
4. none of these

Q.

Find the angles between the lines $\sqrt{3}x+y=1$ and $x+\sqrt{3}y=1$

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

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. 4150
2. 4052
3. 3971
4. 4167

Q.

The angle between any two diagonals of a cube is:

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

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

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

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

Q. A line makes angles $a,b,c,d$ with the four diagonals of a cube, then ${\cos }^{2}a+{\cos }^{2}b+{\cos }^{2}c+{\cos }^{2}d=$

1. $\frac{1}{3}$
2. $\frac{4}{3}$
3. $\frac{2}{3}$
4. $\frac{5}{3}$

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. $\frac{l_1-l_2}{2cos\frac{\theta }{2}},\frac{m_1-m_2}{2cos\frac{\theta }{2}},\frac{n_1-n_2}{2cos\frac{\theta }{2}}$