Cross product Homework help

[ballaholic8]

Hey guys i was having some trouble proving this , can anyone help me with some specific steps i need to understand how to do this question. thanks

 

[galactus]

A cross product is built from determinants.

$\displaystyle \begin{vmatrix}i&j&k\\u_{1}&u_{2}&u_{3}\\1&2&3\end{vmatrix}=i\begin{vmatrix}u_{2}&u_{3}\\2&3\end{vmatrix}-j\begin{vmatrix}u_{1}&u_{3}\\1&3\end{vmatrix}+k\begin{vmatrix}u_{1}&u_{2}\\1&2\end{vmatrix}$

Remember, a 2X2 determinant is $\displaystyle \begin{vmatrix}a_{1}&a_{2}\\b_{1}&b_{2}\end{vmatrix}=a_{1}b_{2}-a_{2}b_{1}$

You can build a system of equations and solve for $\displaystyle u_{1}, \;\ u_{2}, \;\ u_{3}$

 

[skeeter]

let u = <a,b,c>

[i j k]
[a b c] =
[1 2 3]

(3b-2c)i - (3a-c)j + (2a-b)k

3b-2c = 1
3a-c = -1
2a-b = 1

solve for u = <a,b,c>