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