Find all vectors $X =\langle x,y,z \rangle$ for which $\langle 2,1,2\rangle \times \langle x,y,z\rangle = \langle 1,2,-2 \rangle$

Question

This is a homework assignment for Calc 3 and I am kind of stuck. Here is what i've done so far.

$$ \begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 1 & 2 \\ x & y & z \\ \end{matrix} $$ This equals: $$[z-2y]\hat{i} - [2z-2x]\hat{j} + [2y-x]\hat{k} = <1,2,-2>$$

Then I set these equal to their components.

$$ z-2y = 1 \\ 2z-2x =2 \\ 2y-x = -2 \\ $$

From here I tried to solve for one of the variables. $$ 2y-x=-2 \\ +-2y+z = 1\\ $$ From this I got $$z=x-1$$

Plugging this into the second equation I get. $$2(x-1)-2x=2\\ 2x-2-2x=2\\ -2=2 $$

Now I know -2 does not equal 2. So I think I setup my problem wrong or I solved my system of equations wrong.

Thanks in advance for any help and sorry if I have some weird formatting errors i'm new to MathJax.

Answer 1

The problem is I dropped a negative. The second equation should be $−(2z−2x)=2$.