   Chapter 12.5, Problem 83E

Chapter
Section
Textbook Problem

# If a, b, and c are not all 0, show that the equation ax + by + cz + d = 0 represents a plane and ⟨a, b, c⟩ is a normal vector to the plane.Hint: Suppose a ≠ 0 and rewrite the equation in the form a ( x +   d a )  +  b ( y   −  0 )  +  c ( z   −  0 )  = 0

To determine

To show: That the equation ax+by+cz+d=0 represents a plane and a,b,c is a normal vector to the plane.

Explanation

Given data:

If a, b, and c are not all zero, the equation ax+by+cz+d=0 is to be shown as plane equation and the vector a,b,c as normal vector to the plane.

Formula used:

Write the expression to find equation of the plane through the point P0(x0,y0,z0) with normal vector n=a,b,c to the plane as follows.

a(xx0)+b(yy0)+c(zz0)=0 (1)

Case-i: If a0.

Rewrite the equation as ax+by+cz+d=0 follows.

(ax+d)+by+cz=0a[x+da]+b(y0)+c(z0)=0

a[x(da)]+b(y0)+c(z0)=0 (2)

As the equation (2) is similar to the equation (1), the equation ax+by+cz+d=0 is a plane equation through the point (da,0,0) with the normal a,b,c to the plane.

Hence, the equation ax+by+cz+d=0 represents a plane equation with the normal vector a,b,c to the plane when a0.

Case-ii: If b0.

Rewrite the equation as ax+by+cz+d=0 follows

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