   Chapter 12.5, Problem 39E ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# DISCOVER ■ PROVE: Geometric Invariants Do you expect that the distance between two points is invariant under rotation? Prove your answer by comparing the distance d ( P , Q ) and d ( P ′ , Q ′ ) where P ′ and Q ′ are the images of P and Q under a rotation of axes.

To determine

Whether the distance between two points is invariant or not under rotation and the proof for it.

Explanation

Given:

The coordinates P and Q are the original points.

The coordinates P and Q are the images of P and Q under the rotation of axes.

Approach:

A quantity is invariant under rotation if it does not change when the axes is rotated.

The formulas for the rotation of axes is given as,

X=(xcosϕ+ysinϕ)

Y=(xsinϕ+ycosϕ)

Here, X and Y are the points for the image and x and y are the original points.

The distance formula is given as,

d(P,Q)=(x1x2)2+(y1y2)2

Calculation:

The coordinates are given as,

P(x1,y1) and Q(x2,y2).

The coordinates for the images are given as,

P(X1,Y1) and Q(X2,Y2).

Substitute the values of X and Y in the coordinates for the image as,

P(X1,Y1)=P(x1cosϕ+y1sinϕ,x1sinϕ+y1cosϕ)

Q(X2,Y2)=Q(x2cosϕ+y2sinϕ,x2sinϕ+y2cosϕ)

Compute the distance between P and Q using the distance formula as,

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