Home | | Maths 12th Std | Distance between two parallel planes

# Distance between two parallel planes

Mathematics : The distance between two parallel planes

Distance between two parallel planes

### Theorem 6.21

The distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is given by ### Proof

Let A( x1 , y1 , z1 ) be any point on the plane ax + by + cz + d2 = 0 , then we have

ax1 + by1 + cz1 + d2 = 0 ax1 + by1 + cz1 = −d2

The distance of the plane ax + by + cz + d1 = 0 from the point A( x1 , y1 , z1 ) is given by Hence, the distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 given by δ = .

### Example 6.51

Find the distance between the parallel planes x + 2 y − 2z +1 = 0 and 2x + 4 y − 4z + 5 = 0

Solution

We know that the formula for the distance between two parallel planes ax + by + cz + d1 = 0 and ax + by + cz + d2 = 0 is Rewrite the second equation as x + 2y – 2z + 5/2 = 0. Comparing the given equations with the general equations, we get a = 1, b = 2, c = −2, d1=1, d2 = 5/2.

Substituting these values in the formula, we get the distance Example 6.52

Find the distance between the planes Solution

Let be the position vector of an arbitrary point on the plane (2 ˆi − ˆj − 2 ˆk ) = 6 . Then, we have (2 ˆ− ˆ− 2 ˆ) = 6                           .................(1)

If δ is the distance between the given planes, then δ is the perpendicular distance from to the plane Tags : Definition, Theorem, Proof, Solved Example Problems, Solution , 12th Mathematics : UNIT 6 : Applications of Vector Algebra
Study Material, Lecturing Notes, Assignment, Reference, Wiki description explanation, brief detail
12th Mathematics : UNIT 6 : Applications of Vector Algebra : Distance between two parallel planes | Definition, Theorem, Proof, Solved Example Problems, Solution