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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 αx+βy+γz=δare perpendicular if and only if aα+bβ+cγ=0

Step by step solution

01

Given information

The planes determined by the equations ax+by+cz=dand αx+βy+γz=δ

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=⟨a,b,c⟩ and of the equation αx+βy+γz=δ is N2=⟨α,β,γ⟩

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·N2=0

The orthogonal condition N1·N2=0gives:

⟨a,b,c⟩·⟨α,β,γ⟩=0aα+bβ+cγ=0(Dot Product)

Thus, the planes determined by the equations ax+by+cz=dand αx+βy+γz=δ are perpendicular if and only if aα+bβ+cγ=0

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Most popular questions from this chapter

In Exercises 36–41 use the given sets of points to find:

(a) A nonzero vector N perpendicular to the plane determined by the points.

(b) Two unit vectors perpendicular to the plane determined by the points.

(c) The area of the triangle determined by the points.

P(1,4,6),Q(−3,5,0),R(3,2,−1)

Find a vector in the direction of3,1,2and with magnitude 5.

Fill in the blanks to complete each of the following theorem statements:

For ε>0,f(x)∈(L-ε,L+ε)if and only if<ε.

In Exercises 24-27, find compuv,projuv and the component of v orthogonal tou.

u=2,0,-5,v=-3,7,-1

In Exercises 30–35 compute the indicated quantities when u=(−3,1,−4),v=(2,0,5),andw=(1,3,13)

Find the area of the parallelogram determined by the vectors u and v.
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