1. Find the angle between the two planes x  +  2y  +  2z - 5  =  0 and  3x  + 3y  +  2z - 8  = 0

Sol: Given Planes: x  +  2y  +  2z  -  5 =  0  .........( 1 )

3x  + 3y  + 2z  - 8  =  0  ........( 2 )
                   
                    
                  

2. Find the Perpendicular distance of the plane 2x + y + 2z - 17 =  0 from the point (2, 1, 0)

Sol: Given plane  2x + y + 2z  - 17 =  0

Given Point  ( 2,  1,  0 )

The Perpendicular distance from ( x1,  y1,  z1 ) to the plane ax  +  by  +  cz  +  d  =  0  is

3. Find the Equation fo the plane through  ( - 2,  - 1,  - 3 ) and parallel to the plane 2x  +  y  +  z   - 9   =  0

Sol: Given Plane 2x + y + z - 9  =  0

         Given point P ( - 2,  - 1,  - 3 )

Equation of the plane through ( x₁,  y₁,  z₁ ) and parallel to ax + by + cz + d  =  0 is

a ( x - x₁ )  +  b ( y - y₁ )  +  c ( z -z₁ )  =  0

Required plane is  2 ( x + 2 )  +  1 ( y + 1 )  +  1  ( z + 3 )  =  0 

         =  2x  +  y  +  z  +  8  =  0

 ∴   Required plane is 2x  +  y  +  z  +  8  =  0

4. Find the Equation of the plane through ( - 1,  0,  - 6 ) and perpendicular to the line whose direction ratios are ( 6,  20,  - 1 )

Sol: Equation of the plane through ( x₁,  y₁,  z₁ ) and perpendicular to the line whose

direction ratios are ( a,  b,  c  ) is a ( x - x₁ )  +  b  ( y -y₁ )  +  c (  z - z₁ )  =  0

Required plane is  6 ( x + 1 )  + 20  ( y - 0 )  -1  ( z + 6 )  =  0 

                =  6x  +  20y - z  =  0

 ∴  Required plane is  6x  +  20y - z  =  0

5. Find the distance between the parallel planes 2x  +  y - 2z  +  8  =  0 and   2x + y -2z - 19  =  0

Sol: Given parallel planes 2x  +  y  -  2z  +  8  =  0 ......... ( 1 )

                                             2x + y - 2z - 19  =  0 ......... ( 2 )

Distance between the parallel planes ax + by + cz + d₁ = 0,  ax + by + cz + d₂  =  0  is

6. Find the ratio in which the plane 2x + 3y - 2z + y  =  0 divides the line joining the points ( - 1,  2 , 3 ) and ( 2,  3,  5 )

Sol: Given plane 2x + 3y - 2z + 7  =  0

Given points (- 1,  2,  3 ) and  ( 2,  3,  5 )

The ratio in which the plane  π  ≡  ax + by + cz + d  =  0 divides the line segment joint

 A  ( x₁,  y₁,  z₁ ) and B ( x₂,  y₂,  z₂ )

is  –π₁₁₁  :  π₂₂  where  π₁₁₁  =  ax₁ + by₁ + cz₁ + d, and  π₂₂₂  =  ax₂ + by₂ + cz₂+ d

Required ratio =  - π₁₁₁  :  π₂₂₂
          ⇒     - [ 2 ( -1 )  +  3 ( 2 )  -  2 ( 3 )  +  7  ]   :   [  2 ( 2 )  +  3 ( 3 )  -  2 ( 5 )  +  7  ]
         ⇒       - 2 :10
           =   1  :  5   Externally
∴    The plane divides the points externally in  1  :  5  ratio

7. Show that the points ( 2,  3,  4 ),  ( 1,  2,  3 )  lie  in  the  same  side  of  the  plane 3x  -  2y  +  z  -  5   =   0

Sol: Given plane 3x - 2y + z - 5  =  0
Given points A  ( 2,  3,  4 ),  B  ( 1,  2,  3 )
The points A and B lie in the same side (or) Opposite side of the plane π  =  0
According as π₁₁₁ , π₂₂₂ have the same sign (or) oppositic signs.
⇒   π ₁₁₁     =    3 ( 2 ) - 2 ( 3 )  +  4  - 5
⇒   π ₁₁₁     =    6  -  6  +  4  - 5
⇒   π ₁₁₁     =    - 1  <  0
⇒   π ₂₂₂    =    3 ( 1 )  - 2 ( 2 )  + 3  - 5
⇒      π ₂₂₂    =    3 - 4  +  3  - 5
⇒     π ₂₂₂    =    - 3  <  0 
∴     π 111 and π 222 both have same sign given points lie in the same side of plane.

8. Find the equation of the plane having intercepts -4, 5, 6 on the x, y and z axes respectively.

Sol: Given  x  -  intercept :  a  =  -  4
                     y  -   intercopt :  b  =     5
                     z  -    intercopt :  c  =     6
Equation of the plane having x -  intercept  a, y -  intercept  b, z -  intercept  c  is

9. Show that  ( 2,  -3,  1 ) lies in the origin side of  2x + 3y + 4z + 7  =  0

Sol: Given plane  2x + 3y + 4z + 7  =  0
Given point ( 2,  - 3,  1 )

10. Write the equation of the plane 5x + 2y - 3z - 30  =  0 in the intercept form

Sol:  Given Plane 5x + 2y – 3z – 30  =  0
                   5x + 2y – 3z  =  30

11. Reduce the eqation  2x - 2y + z + 6  =  0 of a plane into normal form.

Sol: Given plane  2x - 2y + z + 6 =  0

Writer  -  Anwar