# Determining Motion from a Derivative

## Homework Statement

Given position function r(t) and r'(t) = c X r(t), where c is some constant vector, describe the path of the particle. In other words, describe r(t).

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## The Attempt at a Solution

a) r'(t) points in the direction of motion. If we can understand how r'(t) changes direction, we can understand how r(t) moves.
b) r'(t) is the cross product of c and r(t). Therefore, the function is perpendicular to both c and r(t).
c) If r'(t) is perpendicular to c, r'(t) lies on a plane that c is normal to.

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RUber
Homework Helper
Consider iterating a few times with Newton's method to see what happens.
For starters, assume that C = Identity and evaluate the cross product.

what does C = identity mean, sorry? a constant function?

if c is given by <C1,C2,C3> and I cross that by r(t) <R1,R2,R3>, I get a cross product.:
<C2(R3) - C3(R2), C1(R3) - C3(R1), C1(R2) - C2(R1)>

I then multiply that cross product by time interval t. This gives the position function's next approximate position.

\$<t[C2(R3) - C3(R2)], t[C1(R3) - C3(R1)], t[C1(R2) - C2(R1)]t>

Subsequent iterations become very messy. Are you sure about this? No conceptual way of understanding the problem?

haruspex
Homework Helper
Gold Member
Dot both sides with r(t) and solve. What does this tell you about |r(t)|?

vela
Staff Emeritus
You might find it helpful to choose a coordinate system to make the analysis easier. Orient the coordinate system so that $\vec{c}$ points along the z-axis. That'll give you a much simpler expression for $\vec{c}\times\vec{r}$.