Tag: angle between two planes

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Questions Related to angle between two planes

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

If the planes $ 2x-y+ \lambda z- 5=0$ and $x+4y+2z- 7= 0$ are perpendicular, then $\lambda=$

  1. $1$
  2. $-1$
  3. $2$
  4. $-2$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since, the planes $2x-y+\lambda z-5=0$ & $x+4y+2z-7=0$ are perpendicular to each other
Therefore, $\left( 2i-j+\lambda k \right) .\left( i+4j+2k \right) =0$
$\Rightarrow 2-4+2\lambda =0$
$\Rightarrow \lambda =1$

Ans: A

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

If the planes $\vec{r}. (2\widehat{i}- \widehat{j}+ 2\widehat{k})= 4$ and $\vec{r}. (3\widehat{i}+ 2\widehat{j}+\lambda\widehat{k})= 3$ are perpendicular, then $\lambda =$

  1. $2$
  2. $-2$
  3. $3$
  4. $-3$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Since, the planes $\vec { r } .(2\widehat { i } -\widehat { j } +2\widehat { k } )=4$ & $\vec{r}. (3\widehat{i}+ 2\widehat{j}+\lambda\widehat{k})= 3\ $ are perpendicular to each other
Therefore, $\left( 2i-j+2 k  \right) .\left( 3i+2j+\lambda k \right) =0$
$\Rightarrow 6-2+2\lambda =0$
$\Rightarrow \lambda =-2$

Ans: B

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

The angle between the planes, $\vec{r}.(2\widehat{i}- \widehat{j}+\widehat {k})=6$ and $\vec{r}.(\widehat{i}+ \widehat{j}+2\widehat {k})=5$ , is:

  1. $\dfrac{\pi}{3}$
  2. $\dfrac{2\pi}{3}$
  3. $\dfrac{\pi}{6}$
  4. $\dfrac{5\pi}{6}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Angle between $2x-y+z=6$ & $x+y+2z=5$ is
$\theta =\cos ^{ -1

}{ \left[ \dfrac { \left( 2i-j+k \right) .\left( i+j+2k \right)  }{

\sqrt { \left( { 2 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } \right) \left( 1^{ 2

}+{ 1 }^{ 2 }+2^{ 2 } \right)  }  }  \right]  } =\dfrac { \pi  }{ 3 } $

Ans: B

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

The angle between the planes $ 3x-6y+2z+5=0 $ 7 $ 4x-12y+3z=3 $.Which is bisected by the plane
$ 67x-162y+47z+44 = 0 $is the angle which-

  1. contains origin

  2. is acute

  3. is obtuse

  4. is right angle
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To determine if the bisecting plane contains the origin, check the signs of the expressions for the two given planes at the origin. The bisector that contains the origin is the one where the signs of the constants match the signs of the plane equations.

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

A plane$ P _{1}$ has the equation $2x-y+z=4$ and the plane $P _{2}$ has the equation $x+ny+2z=11.$ If the angle between $P _{1}$ and $P _{2}$ is $\pi /3$ then the value (s) of '$n$' is (are)

  1. $7/2$
  2. $17,-1$
  3. $-17,1$
  4. $-7/2$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
A plane$ P _{1}$ has the equation $2x-y+z=4$ and the plane $P _{2}$ has the equation $x+ny+2z=11.$  
The direction vector of the normal of the first plane is $2i-j+k$ and second plane is $i+nj+2k$.
The angle between $P _{1}$ and $P _{2}$ is $\pi /3$
$\cos { \dfrac { \pi  }{ 3 }  } =\dfrac { \left( 2i-j+k \right) .\left( i+nj+2k \right)  }{ \sqrt { \left( { 2 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } \right) \left( { 1 }^{ 2 }+{ n }^{ 2 }+{ 2 }^{ 2 } \right)  }  } $
$\Rightarrow \dfrac { 1 }{ 2 } =\dfrac { 4-n }{ \sqrt { 6\left( 5+{ n }^{ 2 } \right)  }  } $
$\Rightarrow \cos^2(\dfrac{\pi}{3})=\dfrac{(4-n)^2}{6(5+n^2)} $
$\Rightarrow n^2+16n-17=0\Rightarrow (n+17)(n-1)=0$
$\Rightarrow n=-17,1$
Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

The angle between the planes $\displaystyle x + y + z = 0$ and $\displaystyle 3x - 4y + 5z = 0$ is

  1. $\displaystyle \cos ^{-1}\left ( \frac{1}{5} \sqrt{\frac{2}{5}} \right )$
  2. $\displaystyle \frac{\pi }{2}$
  3. $\displaystyle \frac{\pi }{3}$
  4. $\displaystyle \cos ^{-1}\left ( \frac{2}{5} \sqrt{\frac{2}{3}} \right )$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The angle between $x+y+z=0$ & $3x-4y+5z=0$ is
$\theta =\cos ^{ -1

}{ \left[ \dfrac { \left(\vec  i+\vec j+\vec k \right) .\left( 3\vec i-4\vec j+5\vec k \right)  }{

\sqrt { \left( { 1 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } \right) \left( 3^{ 2

}+{ 4 }^{ 2 }+5^{ 2 } \right)  }  }  \right]  } $

$= \cos ^{ -1 } \left( \dfrac { 2 }{ 5 } \sqrt { \dfrac { 2 }{ 3 }  }  \right) $

Ans: D

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

Which of the following planes is equally inclined to the planes $\displaystyle 4x + 3y - 5z = 0$ and $\displaystyle 5x - 12y + 13z = 0$?

  1. $\displaystyle 11x - 3y = 0$
  2. $\displaystyle 3x + 11y = 0$
  3. $\displaystyle 3x + 11y = 65z$

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The bisector of planes $4x+3y-5z=0$ & $5x-12y+13z=0$ can be given by

$\dfrac { 4x+3y-5z }{ \sqrt { { 4 }^{ 2 }+{ 3 }^{ 2 }+{ 5 }^{ 2 } }  } \pm \dfrac { 5x-12y+13z }{ \sqrt { { 5 }^{ 2 }+12^{ 2 }+13^{ 2 } }  } =0$

$\Rightarrow 52x+39y-65z\pm \left( 25x-60y+65z \right) =0$

$\Rightarrow 11x-3y=0$ and $27x+99y-130z=0$

Ans: A

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

The equation of the plane bisecting the acute angle between the planes $\displaystyle x - y + z - 1 = 0$ and $\displaystyle x + y + z = 2$ is

  1. $\displaystyle x + z = \frac{3}{2}$
  2. $\displaystyle 2y = 1$
  3. $\displaystyle x - y - z = 3$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Given planes are  $ x-y+z-1=0.....(1)$ and $x+y+z-2=0.........(2)$
Therefore equation of plane bisecting these planes are
$\dfrac{x-y+z-1}{\sqrt{3}}=\pm\dfrac{x+y+z-2}{\sqrt{3}}$
$\Rightarrow x+z = \dfrac{3}{2}.......(3)$ and $y = \dfrac{1}{2}.......(4)$
If $\theta$ is the angle between (2) and (4) then,
$  \cos\theta = \dfrac{1/2}{(1/2).(\sqrt{3})}=\dfrac{1}{\sqrt{3}}$
$\Rightarrow \theta > 45^\circ$
Hence plane (4) bisects the obtuse angle between the given planes.
Therefore equation of plane bisecting acute angle  between given plane is
$x+z = \dfrac{3}{2}$

Multiple choice maths the plane angle between planes angle between two planes problems involving equation of plane

The angle between the planes $\bar { r } \cdot \bar { n _{ 1 } } =\left| \bar { { d } _{ 1 } }  \right| $ and $\bar { r } \cdot \bar { n _{ 2 } } =\left| \bar { { d } _{ 2 } }  \right| $

  1. $\cos^{-1}\left(\displaystyle \frac{\bar{n _{1} }\cdot\bar{d} _{1}}{\left | \bar{d} _{1}\times \bar{d} _{2} \right |}\right)$
  2. $\cos^{-1}\left(\displaystyle \frac{\bar{n} _{1}.\bar{n} _{2}}{\left |\bar{n} _{1} \right |\left | \bar{n} _{2} \right |}\right)$
  3. $\cos^{-1}\left(\displaystyle \frac{\bar{n} _{1}\bar{n} _{2}}{\bar{n} _{1}\times \bar{n} _{2} }\right)$
  4. $\cos^{-1}\left(\displaystyle \frac{\bar{n} _{1}\cdot \left | \bar{d} _{2} \right |}{\left | \bar{n} _{1} \right |\left | \bar{n} _{2} \right |}\right)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Given planes are $\bar { r } \cdot \bar { n _{ 1 } } =\left| \bar { { d } _{ 1 } }  \right| $ and $\bar { r } \cdot \bar { n _{ 2 } } =\left| \bar { { d } _{ 2 } }  \right| $ 

Angle between the planes is same as the angle between the normal vectors.
Hence the angle  $\theta=\cos^{-1}\left(\dfrac{\bar{n} _1.\bar{n} _2}{|\bar{n} _1||\bar{n} _2|}\right)$