Distance between two parallel planes
The distance between two parallel planes ax + by + cz
+ d1 = 0 and ax + by + cz + d2
= 0 is given by
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 = .
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 â‹…
Solution
Let be the position vector of an arbitrary
point on the plane ⋅(2 ˆi − ˆj −
2 ˆk ) = 6 .
Then, we have
⋅(2 ˆi − ˆj − 2 ˆk ) = 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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