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before going to bed :) Particle P satisfies the differential equation\[ \frac{\text{d}\mathbf{r}}{\text{d}t}=\mathbf{c}\times \mathbf{r}\]show that P moves with a constant speed on a circular path. where c is a constant vector

Mathematics
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If c is constant then cross-product unnecessary, right? It's a scalar multiple?
I'm tempted to say experiment, as electrons do this (Lorentz force), even though it's second derivatives involved the result looks very similar. It's not the answer, but it's an interesting analogy.
And obviously it wouldn't always be circular, unless it starts of perpendicular, that would be circular if you looked parallel to the c-feild. And if it starts parallel, then it's just a straight line, but I'm avoiding the question.

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