Magnitudes are positive values, but when I take, for example: the magnitude of a position vector: $r = 3 - 0.04t^2$ and try to take the derivate of it, the result will be $v = -2 * 0.04t$ which is a negative quantity. I don't really see the mistake here but that negative sign should not be there. Can a rate of change be negative in this case?
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1$\begingroup$ The norm/magnitude of a vector, $||\mathbf{X}||=\sqrt{\mathbf{X}\cdot \mathbf{X}}$, is always positive, but the components of a vector can be negative, is that what your mixing up maybe? $\endgroup$– user330563Commented Aug 27, 2022 at 7:48
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2 Answers
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Yes, a magnitude is positive, but the derivative of a magnitude may be negative. For example, speed is the magnitude of velocity, so speed is positive. But when I apply my car’s brakes the derivative of my speed is negative.
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$\begingroup$ Isn't counterintuitive that the derivate of a magnitude is not a magnitude too? In this case the negative sign only indicates decrease but not direction right? $\endgroup$– GaboCommented Aug 28, 2022 at 3:49
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Here velocity is negative as its difference between two coordinates (final-initial) so it's velocity's direction which is negative in x axis and if you want magnitude than simply do mod on both sides and so u get its magnitude which is positive
