Prove two vectors are collinear using ad=bc

Question

Suppose $\vec{u}$ and $\vec{v}$ span $\mathbb{R}^2$. if $\vec{x}=a\vec{u}+b\vec{v}$ and $\vec{y}=c\vec{u}+d\vec{v}$ are colinear, prove that $ad=bc$.

Answer 1

Collinearity of $2$ vectors means that one vector is a constant times the other, i.e $\vec y = k\vec x\implies (c,d) = k(a,b)\implies c = ka, d = kb\implies \dfrac{c}{a}= \dfrac{d}{b} = k\implies ad = bc$ .