   Chapter 12.5, Problem 92E ### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

#### Solutions

Chapter
Section ### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Assume that the elliptical orbit r = e d 1 + e   cos   θ is in the xy-plane, with L along the z-axis. Prove that | | L | | = r 2 d θ d t .

To determine

To Prove: The vector L=r2dθdt.

Explanation

Given:

The elliptical orbit r=ed1+ecosθ, L is along the z-axis.

Proof:

Let r(t)=rcosθi+rsinθj be the position vector.

Differentiate the above expression with respect to θ;

r(t)=ddt(rcosθi+rsinθj)r(t)=rsinθdθdti+rcosθdθdtj

Then, take the cross product of r(t) and r(t);

r×r=|ijkrcosθrsinθ0rsinθ

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