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Definition, Theorem, Proof, Solved Example Problems, Solution - Distance between two parallel planes | 12th Mathematics : UNIT 6 : Applications of Vector Algebra

Chapter: 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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


 

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12th Mathematics : UNIT 6 : Applications of Vector Algebra : Distance between two parallel planes | Definition, Theorem, Proof, Solved Example Problems, Solution

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12th Mathematics : UNIT 6 : Applications of Vector Algebra


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