Mathematics : The distance between two parallel planes

**Distance between two parallel planes**

The distance between two parallel planes *ax *+ *by *+ *cz
*+ *d*_{1} = 0 and *ax *+ *by *+ *cz *+ *d*_{2}
= 0 is given by

Let A( *x*_{1} , *y*_{1} , *z*_{1}
) be any point on the plane *ax *+ *by *+ *cz *+ *d*_{2}
= 0 , then we have

*ax*_{1} + *by*_{1} + *cz*_{1} + *d*_{2}
= 0 â‡’ *ax*_{1} + *by*_{1} + *cz*_{1}
= âˆ’*d*_{2}

The distance of the plane *ax *+ *by *+ *cz *+ *d*_{1}
= 0 from the point *A*( *x*_{1} , *y*_{1} , *z*_{1}
) is given by

Hence, the distance between two parallel planes *ax + by + cz + d _{1}* = 0 and

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

**Solution**

We know that the formula for the distance between two parallel
planes *ax *+ *by *+ *cz *+ *d*_{1} = 0 and *ax *+
*by *+ *cz *+ *d*_{2} = 0 is Rewrite the second
equation as *x* + 2*y* â€“ 2*z* + 5/2 = 0.
Comparing the given equations with the general equations, we get *a *= 1, *b
*= 2, *c *= âˆ’2, *d*_{1}=1, *d*_{2} = 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 â‹…(2 Ë†*i *âˆ’ Ë†*j *âˆ’
2 Ë†*k *) = 6 .
Then, we have

â‹…(2 Ë†*i *âˆ’ Ë†*j *âˆ’ 2 Ë†*k *) = 6 .................(1)

If *Î´ *is the distance between the given planes, then *Î´ *is
the perpendicular distance from to the plane

Tags : Definition, Theorem, Proof, Solved Example Problems, Solution , 12th Mathematics : UNIT 6 : Applications of Vector Algebra

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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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