   Chapter 11, Problem 18PS

Chapter
Section
Textbook Problem

# Distance Between Parallel PlanesShow that the distance between the parallel planes a x + b y + c z + d 1 = 0 and a x + b y + c z + d 2 = 0 is Distance = | d 1 − d 2 | a 2 + b 2 + c 2

To determine

To prove: The distance between the parallel planes ax+by+cz+d1=0 and ax+by+cz+d2=0 is:

|d1d2|a2+b2+c2

Explanation

Given:

Two parallel planes ax+by+cz+d1=0 and ax+by+cz+d2=0.

Formula used:

Projection of v onto n is |v·n|n||.

Proof:

Let there be two arbitrary points P1(x1,y1,z1) and P2(x2,y2,z2) on planes ax+by+cz+d1=0 and ax+by+cz+d2=0 respectively.

Then there can be a vector v=x2-x1,y2-y1,z2-z1 starting at point P1 and ending at P2.

The distance between the two planes can be computed by the projection of vector v onto the normal vector n=a,b,c.

Projection of v onto n is |v·n|n||

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