Question

use properties of determinants (that is, don't use a direct computational verification) to show that for any 3-dimensional vectors u and v, u x v = - v x u

Answer 1

Do you know how to find cross products using a determinant of a matrix?

Answer 2

yes but we aren't allowed to use a direct computational method. so i'm thinking explaining it using laplaces expansion theorem. what do you think?

Answer 3

Instead of using direct computation, just let \( u = <u_1, u_2, u_3>, \quad v=<v_1, v_2, v_3> \)and solve \(u\times v\) and \(-v \times u\). Or does your teacher count this as direct computation?

Answer 4

oh ignore my last comment, that was for a different question similiar to this one. and i'm thinking that he counts that as a "direct computation"

Answer 5

Then at least write out the determinants.

Answer 6

Then you should get \[u \times v=\left[\begin{matrix} i&j&k\\u_1&u_2&u_3\\v_1&v_2&v_3 \end{matrix}\right]\]\[-v \times u=\left[\begin{matrix} i&j&k\\-v_1&-v_2&-v_3\\u_1&u_2&u_3 \end{matrix}\right]\]

Answer 7

After that, I'm not positive, but you've probably learned some properties of determinants that might be able to help you.

Answer 8

thanks

Answer 9

You're welcome.

Answer 10

If all else fails, just solve the determinants, and do it that way.