How Do I Find the Derivative of This Vector Function?

[megr_ftw]

Homework Statement

Find the derivative of the vector function
r(t)=ta X (b+at)
where a=<4,5,2>, b=<1,-3,2>, and c=<4,3,1>

Homework Equations

The Attempt at a Solution

I know how to take the derivative and everything but the way this question is worded confuses me!
I'm assuming the X means cross product? but it may just mean multiply. Do I plug in the values of a,b,c, and then do what with all the t's?

 

[Mark44]

I'm pretty sure X means cross product. People generally don't use X to mean ordinary multiplication at the calculus level.

Yes, substitute the values for a, b, and c, and then carry out the cross product. You'll end up with either (...)i + (...)j + (...)k or <..., ..., ...>, both of which will have terms with t in them. To get r'(t), just take the derivative of each of the three components.

 

[megr_ftw]

should I distribute the t to the a values?

 

[Mark44]

Yes. t is a scalar, so ta = <4t, 5t, 2t>. at is the same as ta.

 

[HallsofIvy]

Don't carry out the product! Just use the product rule for vector multiplication:
$\vec{f}= \vec{a}t\times (\vec{b}+\vec{c}t$)

so $\vec{f}'= \vec{a}\times (\vec{b}+ \vec{c}t)+ \vec{a}t \times \vec{c}= \vec{a}\times\vec{b}+ 2\vec{a}\times\vec{c}t$.