Prove linear dependence of 3 vectors

Question

I am trying to prove linear dependence, but so far i have not been successfull. Could you point me in the right the direction?

For all $\vec{a},\vec{b},\vec{c},x,y,z$ prove that vectors $x\vec{a}-y\vec{b},z\vec{b}-x\vec{c},y\vec{c}-z\vec{a}$ are lineary dependent.

Answer 1

If $x,y,z$ are all zero then the result is obvious. Otherwise:-

Multiply your 3 vectors by $z,y$ and $x$, respectively. This gives zero and so the three vectors are linearly dependent,