Chapter 12.5, Problem 92E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

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+ecosÎ¸, L is along the z-axis.

Proof:

Let r(t)=rcosÎ¸i+rsinÎ¸j be the position vector.

Differentiate the above expression with respect to Î¸;

râ€²(t)=ddt(rcosÎ¸i+rsinÎ¸j)râ€²(t)=âˆ’rsinÎ¸dÎ¸dti+rcosÎ¸dÎ¸dtj

Then, take the cross product of r(t)Â andÂ râ€²(t);

rÃ—râ€²=|ijkrcosÎ¸rsinÎ¸0âˆ’rsinÎ¸

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