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Chapter 10: Q 16. (page 845)

Explain how to tell when two planes are perpendicular.

Short Answer

Expert verified

The planes determined by the equations ax+by+cz=dand 伪x+尾y+纬z=未are perpendicular if and only if a伪+b尾+c纬=0

Step by step solution

01

Given information

The planes determined by the equations ax+by+cz=dand 伪x+尾y+纬z=未

02

Calculation

The goal is to demonstrate how to detect if two planes are perpendicular to one another.

The normal vector of the equation ax+by+cz=d is N1=鉄�a,b,c鉄� and of the equation 伪x+尾y+纬z=未 is N2=鉄�伪,尾,纬鉄�

03

Calculation

Only if and only if the planes' normal vectors are orthogonal are they considered orthogonal.

The normal vectors are orthogonal if and only if N1路N2=0

The orthogonal condition N1路N2=0gives:

鉄�a,b,c鉄�路鉄�伪,尾,纬鉄�=0a伪+b尾+c纬=0(Dot Product)

Thus, the planes determined by the equations ax+by+cz=dand 伪x+尾y+纬z=未 are perpendicular if and only if a伪+b尾+c纬=0

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