Angle between Two Planes

Q. The equation of the plane passing through the points (1, -3, -2) and perpendicular to planes x +2y+2z=5 and 3x+3y+2z=8, is

1. 2x -4y +3z-8 =0
2. 2x-4y-3z+8 =0
3. 2x+4y+3z+8 =0
4. x-3y+z-5=0

Q. The equation of the plane passing through the points (1, -3, -2) and perpendicular to planes x +2y+2z=5 and 3x+3y+2z=8, is

1. 2x -4y +3z-8 =0
2. 2x-4y-3z+8 =0
3. 2x+4y+3z+8 =0
4. x-3y+z-5=0

Q. Angle between two planes $a_{1}x+b_{1}x+c_{1}x+d_1=0$ & $a_{2}x+b_{2}x+c_{2}x+d_2=0$ is given by-

1. $tan\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
2. $sin\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
3. $cos\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
4. $cot\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$

Q. The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$ are :

1. $\sqrt{2},1,-1$
2. $2\sqrt{3},1,-1$
3. $2,-1,1$
4. $2,\sqrt{2},-\sqrt{2}$

Q.

Find the angle between the planes
$2x+7y+11z-3=0\text{and}5x+3y+9z+1=0$

1. ${\cos }^{-1}\left(\frac{130}{\sqrt{174}.\sqrt{115}}\right)$

2. ${\cos }^{-1}\left(\frac{139}{\sqrt{174}.\sqrt{115}}\right)$

3. ${\cos }^{-1}\left(\frac{123}{\sqrt{174}.\sqrt{115}}\right)$

4. ${\cos }^{-1}\left(\frac{107}{\sqrt{174}.\sqrt{115}}\right)$

Q.

Find the angle between planes 2x +7y +11z - 3 = 0 & 5x +3y +9z +1 = 0

1. $co{s}^{-1}=\frac{130}{\sqrt{174}.\sqrt{115}}$
2. $co{s}^{-1}=\frac{139}{\sqrt{174}.\sqrt{115}}$
3. $co{s}^{-1}=\frac{123}{\sqrt{174}.\sqrt{115}}$
4. $co{s}^{-1}=\frac{107}{\sqrt{174}.\sqrt{115}}$

Q. Find the planes bisecting the angle between the planes x + 2y + 2z = 9
and 4x – 3y + 12z + 13 = 0.

1. 25x+17y+62z-78=0
2. 25x+17y+62z+78=0
3. x+35y-10z+156=0
4. x+35y-10z-156=0

Q. The direction ratios of normal to the plane through the points $(0,-1,0)$ and $(0,0,1)$ and making an angle $\frac{\pi }{4}$ with the plane $y-z+5=0$ are :

1. $\sqrt{2},1,-1$
2. $2\sqrt{3},1,-1$
3. $2,-1,1$
4. $2,\sqrt{2},-\sqrt{2}$

Q. In order to find the dip of oil bed below the surface of the ground, vertical boring are made from angular points $A,B,C$ of a $△ABC$ which is in a horizontal plane. Let the depth of oil bed at three points $A,B,C$ are found to be $l,$ $l+k$ and $l+m$ $(k<m)$ respectively. The length of the sides $CA$ and $AB$ are $b$ and $c$ respectively and the angle between them is $A$. If the angle of the dip with horizontal is $\theta$, then

1. $\tan \theta \sin A=\sqrt{\frac{k^2}{c^2}+\frac{m^2}{b^2}-\frac{2km}{bc}\cos A}$
2. $\tan \theta \sin A=\sqrt{\frac{k^2}{b^2}+\frac{m^2}{c^2}-\frac{2km}{bc}\cos A}$
3. Normal to the oil bed plane is $\overrightarrow{n}=bm\hat{i}+ck\hat{j}+bc\hat{k}$, when $A=\frac{\pi }{2}$
4. Normal to the oil bed plane is $\overrightarrow{n}=mc\hat{i}+bk\hat{j}+bc\hat{k}$, when $A=\frac{\pi }{2}$

Q. What is the angle between two planes having normal vectors as
$\overrightarrow{n_1}=\hat{i}+2\hat{j}+2\hat{k}$ and $\overrightarrow{n_2}=4\hat{i}-4\hat{j}+2\hat{k}$ ?

1. 
   $0^o$
2. 
   ${90}^{\circ }$
3. 
   ${45}^{\circ }$
4. 
   ${60}^{\circ }$

Q. Direction ratios of the normal to the plane which passes through the point $(1,0,0)$ and $(0,1,0)$ and makes an angle $\frac{\pi }{4}$ with the plane $x+y=3$ are

1. $(1,1,\sqrt{2})$
2. $(1,1,-\frac{1}{\sqrt{2}})$
3. $(1,1,\frac{1}{\sqrt{2}})$
4. $(1,1,0)$

Q. Angle between two planes $a_{1}x+b_{1}x+c_{1}x+d_1=0$ & $a_{2}x+b_{2}x+c_{2}x+d_2=0$ is given by-

1. $tan\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
2. $sin\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
3. $cos\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$
4. $cot\left(\theta \right)=\frac{a_1{a}_2+b_1{b}_2+c_1{c}_2}{\sqrt{(a_1^2+b_1^2+c_1^2)(a_2^2+b_2^2+c_2^2)}}$

Q. Let $P_1$ denote the equation of a plane to which the vector $(\hat{i}+\hat{j})$ is normal and which contains the line whose equation is $\overrightarrow{r}=\hat{i}+\hat{j}+\hat{k}+\lambda (\hat{i}-\hat{j}-\hat{k})$ and $P_2$ denote the equation of the plane containing the line $L$ and a point with position vector $\hat{j}$. Which of the following holds good?

1. The equation of $P_1$ is $x+y=2$
2. The equation of $P_2$ is $\overrightarrow{r}.(\hat{i}-2\hat{j}+\hat{k})=2$
3. The acute angle between $P_1$ and $P_2$ is ${\cot }^{-1}\left(\sqrt{3}\right)$
4. The angle between the plane $P_2$ and the line $L$ is ${\tan }^{-1}\sqrt{3}$