Vector Functions, Differentiation

[Falcon]

This doesn't exactly have to do with the title, but generally comes up in this sort of thing. I'm feeling pretty dense right now to not be able to understand what's going on.. but my textbook isn't exactly helping.

lets say there is a function which relates a particles position, r, with respect to time, t: (w is representative of the period)

r=3cos(wt)i + 4cos(wt)j + 5sin(wt)k

its derivative should give you its velocity

v=-3wsin(wt)i - 4wsin(wt)j + 5wcos(wt)k

the magnitude of the velocity vector should give the speed of the particle

s= ?? (the book suggests the answer is s=5w)


How does one arrive at the magnitude of vector v?? I can do this in a case where we are looking at simple coefficients of i,j, and k... but with the trig in there, I'm getting sort of confused. If someone would post up a step-by-step, it would be much appreciated!

Thanks! Sorry for asking such a simple question.. probably goes back to grade 10 math that I've just forgotten! lol

 

[Chen]

The vector v can also be represented as (-3wsin(wt), -4wsin(wt), 5wcos(wt)). Use the expanded pythagoras theorem:
|v|² = [-3wsin(wt)]² + [-4wsin(wt)]² + [5wcos(wt)]²
|v|² = 9w²sin²(wt) + 16w²sin²(wt) + 25w²cos²(wt)
|v|² = w²[9sin²(wt) + 9cos²(wt) + 16sin²(wt) + 16cos²(wt)]
|v|² = 25w²
|v| = 5w

If you are not familiar with this method you can use the dot-product:
|v|² = v(dot)v
You basically reach the same expression after multiplying the vector by itself.

 

[Falcon]

Thanks Chen,

I probably should have shown the steps that I had come up with.

I can reach this:
|v|^2 = w^2[9sin^2(wt) + 9cos^2(wt) + 16sin^2(wt) + 16cos^2(wt)]

and i can see how we get the w^2 in the next line
|v|2 = 25w^2

but how do we get the 25 from this mess: [9sin^2(wt) + 9cos^2(wt) + 16sin^2(wt) + 16cos^2(wt)] ??


(btw, how did you manage superscript without tex?)

 

[Falcon]

ahh... OK.. I just got an idea.. good old trig identities.. let me see if I can work it out

EDIT: ahhh... makes perfect sense now :D

9sin^2(wt) + 9cos^2(wt) + 16sin^2(wt) + 16cos^2(wt)

=25sin^2(wt) + 25cos^2(wt)

=25(sin^2(wt) + cos^2(wt))

=25(1)

=25

 

[Chen]

Hint:
sin²(x) + cos²(x) = 1
See, you figured it our yourself.

(Use [ sup ] and [ sub ] tags, without the spaces.)

 

[Chen]

Heh, I'm not really sure why I split it to two pairs of 9 and 16, I could've used just one pair of 25. Old habit I guess.

 

[Falcon]

Yeah, in fact looking at it now, I think I would have just added them all to

|v|² = 25w²sin²(wt) + 25w²cos²(wt)

then factored 25w² all at once