   Chapter 10.4, Problem 54E

Chapter
Section
Textbook Problem

# Distance Formula(a) Verify that the Distance Formula for the distance betweenthe two points ( r 1 , θ 1 ) and ( r 2 , θ 2 ) in polar coordinates is d = r 1 2 + r 2 2 − 2 r 1 r 2 cos ( θ 1 − θ 2 ) .(b) Describe the positions of the points relative to each other for θ 1 = θ 2 , Simplify the Distance Formula for this case. Is the simplification what you expected? Explain.(c) Simplify the Distance Formula for θ 1 − θ 2 = 90 ∘ . Is the simplification what you expected? Explain.(d) Choose two points on the polar coordinate system and find the distance between them. Then choose differentpolar representations of the same two points and apply the Distance Formula again. Discuss the result.

(a)

To determine

To prove: That the distance formula for the distance between the two points given as, (r1,θ1) and (r2,θ2) in terms of polar coordinates is d=r12+r222r1r2cos(θ1θ2).

Explanation

Given:

The given polar coordinates are (r1,θ1) and (r2,θ2).

Formula used:

The formula to calculate the distance between two points d=(x2x1)2+(y2y1)2.

Proof:

Polar coordinates are given as, (r1,θ1) and (r2,θ2). Converts it into Cartesian co-ordinates, (x1,y1)=(r1cosθ1,r1sinθ1) and (x2,y2)=(r2cosθ2,r2sinθ2).

Now, apply the distance formula between the points as shown below,

d=(x2x1)2+(y2y1)2

Substitute the values into the above equation,

d=(r2cosθ2r1cosθ1)2+(r2sinθ2r1sinθ1)2=r12(cosθ1)2+r22(cosθ2

(b)

To determine

To calculate: The position of points relative to each other for θ1=θ2. Also, simplify the distance formula for it..

(c)

To determine

To calculate: The distance formula for (θ1θ2)=90°. Also, explain the result.

(d)

To determine

To calculate: The distance between any two arbitrary points. Also, discuss their result.

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