Vector Differentiation

[mammothrob]

some vector (<cost, sint>)/ t

to take the derivative of this, does the normal quotiet rule apple

[(t)dx<cost,sint> - <cost, sint> dx(t)]/ t^2



or is the rule different becuase the the top is a vector function?

 

[pka]

One must use \(\displaystyle t\) as a scalar in this question.
Thus we get:
\(\displaystyle \L\begin{array}{rcl}
\frac{{\left\langle {\cos (t),\sin (t)} \right\rangle }}{t} & = & \left\langle {\frac{{\cos (t)}}{t},\frac{{\sin (t)}}{t}} \right\rangle \\
\frac{d}{{dt}}\left\langle {\frac{{\cos (t)}}{t},\frac{{\sin (t)}}{t}} \right\rangle & = & \left\langle {\frac{{ - t\sin (t) - \cos (t)}}{{t^2 }},\frac{{t\cos (t) - \sin (t)}}{{t^2 }}} \right\rangle \\
& = & \frac{{t\left\langle { - \sin (t),\cos (t)} \right\rangle - \left\langle {\cos (t),\sin (t)} \right\rangle }}{{t^2 }} \\
\end{array}.\)

 

[mammothrob]

one more thing about this

on your last step when you factored a t from the vector why did it separate it into one vector subtracted from the other? your very last step. thanks

 

[pka]

Re: one more thing about this

on your last step when you factored a t from the vector why did it separate it into one vector subtracted from the other?

Well, it is correct! Is it not?