Why does d/dx[r(t) · r(t)] equal 2r'(t) · r(t)?

[rad0786]

Hey... i was hopeing somebody can help me with a homework question... its about vector calc... taking the deriviative.

d/dx[r(t) dot r(t)] = r'(t) dot r(t) + r(t) dot r'(t) = 2r'(t) dot r(t)

I know it sounds sillly, but i was just wondering how on Earth they got
2r'(t) dot r(t) ?

 

[Hurkyl]

Well, go through the list of properties of the dot product -- any of them look useful?

 

[robphy]

Another approach... what must be true for that last equality
r'(t) dot r(t) + r(t) dot r'(t) = 2r'(t) dot r(t)
to happen (assuming that each side is correct)?
This should lead to the property hinted by Hurkyl.

 

[rad0786]

Okay i see it now.

It comes from the property...

a dot (b + c) = a dot b + a dot c

:)

Thanks for poinitng that out...

 

[Hurkyl]

Ah, right -- that one's so fundamental I forgot it even needed to be invoked! (I had only noticed you needed a.b = b.a)