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Chapter 10: Q 4. (page 845)

Explain why two planes orthogonal to the same vector are either parallel or identical.

Short Answer

Expert verified

The two planes orthogonal to the same vector are either parallel or identical.

Step by step solution

01

Given information

Consider the two planes with normal vectors N1 and N2 orthogonal to the same non-zero vector v

02

Calculation

The goal is to show that two orthogonal planes facing the same vector are either parallel or identical.

Use the fact that the dot product of orthogonal vectors is zero to verify the point.

The plane with normal vectors N1is orthogonal to the vector v

Therefore,

N1·v=0......(1)

The plane with normal vectors N2 is orthogonal to the vector v

Therefore,

N2·v=0……(2)
03

Calculation

From equations (1) and (2) the following result is obtained.

N1·v=N2·vN1·v-N2·v=0N1-N2·v=0(Dot product is associative)

Because vector vis is non-zero.

Therefore,

N1-N2=0N1=N2

Therefore, the two planes orthogonal to the same vector are either parallel or identical.

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Most popular questions from this chapter

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

u×wandw×u

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(3,1,8),Q(0,6,−1),R(−3,5,−3)

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

role="math" localid="1649434688557" v×wandw×v

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.

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

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

u=3,-1,2,v=-4,-6,3
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