How do you prove that a=-(w^2)

[asdf1]

how do you prove that a=-(w^2)
from v=wxr?
i know you're supposed to differentiate v=wxr, but i don't know how to differentiate a cross product...

 

[scholzie]

The cross product is the vector created by the determinant
$$a \times b = <br /> \begin{tabular}{|c c c|}<br /> i & j & k \\ <br /> a_1 & a_2 & a_3 \\<br /> b_1 & b_2 & b_3 \\<br /> \end{tabular}$$

So, take the determinant of the above matrix, and then differentiate as normal. (Hint: the determinant will give you a vector, with 3 coordinates. You can differentiate each coordinate on its own.)

Edit: are you familiar with how to take a determinant? Otherwise you can use $a \times b = |a||b|\sin{\theta}$, but then you'll have to know $\theta$, or treat it as a constant.

 

[asdf1]

wow!
thank you very much!
what happens if you want to prove it for an infinite number of coordinates(ex., i, j, k, l, m, ...)?

 

[HallsofIvy]

wow!
thank you very much!
what happens if you want to prove it for an infinite number of coordinates(ex., i, j, k, l, m, ...)?

The cross product is only defined in R³.