What are the extreme values of the speed for a given parametric curve?

[FrogPad]

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First question:
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This first question is kind of weird. I'm not even sure where to go with it. If anyone has a hint, that would be awesome.

From \vec A \times \vec B = -\vec B \times \vec A deduce \vec A \times \vec A = 0

Can it be as simple as:
let \vec B = \vec A_0 | \vec A_0 = \vec A
thus: \vec A \times \vec A_0 = -\vec A_0 \times \vec A = 0

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Second question:
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Find the minimum and maximum speed if x=t+\cos t, y=t-\sin t.

Please allow me to take advantage of the inner space operator for sake of ease while writing the vectors :)

Thus:
\vec x = <t+\cos t,t-\sin t>
\vec v = <1-\sin t, 1-cos t>

So speed is computed as: |\vec v|. Therefore the largest speed values that can occur are when: \vec v = <1,2> or <2,1> and the lowest speed values that can occur are when \vec v = <1,0> or <0,1>.

Is this reasoning even correct with this problem?

 

[StatusX]

For the first one, I'm not sure what A₀ is, but all you have to do is let B=A in the first identity. For the second, you might want to be a little more rigorous. You can find the extrema of the speed by:

\frac{d}{dt} |\vec v|^2 = 0

And use:\frac{d}{dt} |\vec v|^2 = \frac{d}{dt} (\vec v\cdot \vec v)

Which will lead you to the equation:

\vec v \cdot \vec a = 0

 

[FrogPad]

With the \vec A_0 = \vec A I was just trying to show that I was plugging \vec A into the expression. It really wasn't necessary and actually more confusing (I left this out of the homework).

For the second one. Cool Thank you. That's what I wasn't doing. I needed to find the extrema of the speed not of the velocity vector. I appreciate it.