I am trying to understand dot and cross products from a physics perspective. If a space curve r(t) satisfies the equation r′(t) xx r′′(t)=0 for all t, I understand that the derivative of r′(t) is parallel to r′(t), so the velocity vector r′(t) does not change direction. Thus, this curve moves along a line.

dripcima24 2022-10-02 Answered
I am trying to understand dot and cross products from a physics perspective. If a space curve r(t) satisfies the equation r ( t ) × r ( t ) = 0 for all t, I understand that the derivative of r′(t) is parallel to r′(t), so the velocity vector r′(t) does not change direction. Thus, this curve moves along a line.
However, I'm not sure what r ( t ) r ( t ) = 0 means. My intuition is that it represents motion along a circle, or part of a circle, since the velocity and acceleration vectors should be perpendicular in that case.
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Answers (2)

Haylee Branch
Answered 2022-10-03 Author has 7 answers
Observe the identity
d d t | r ( t ) | 2 = 2 r ( t ) r ( t )
Thus, r ( t ) r ( t ) 0 if and only if the motion r has constant speed.
P.S. The circle you mentioned is somewhat relevant to this, since the velocity vector r′(t) would lie on a fixed circle centered at the origin.
P.S.(again) Actually, r ( t ) × r ( t ) 0 does not imply that r(t) lies on a fixed line in general. You may consider r that stalls at a point, stays at that point for a while(say 1 second), then changes direction, and re-accelerates. Still, your intuition is correct if we add condition that r′(t) is never zero. (the proof is clear; the notion of direction you mentioned now makes sense)
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Drew Williamson
Answered 2022-10-04 Author has 2 answers
You can integrate both sides:
t = 0 t = t f r r ˙ d t = 0
But, d r r d t = 2 ( r r ˙ )
[ r r ] t = 0 t = t f = 0
Or,
r r | t f = r r | t = 0 = C
Meaning that the length of vector is constant. If the length of vector is constant, we can imagine it to be spinning about a origin. A physics example would be to keep origin at the center of a circle and track how a particle spins around a circular loop centered at the same origin.
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