I have absolutely no clue. Can someone at least get me started?
Thanks a Million!!!
Hello qbkr21For the geometric argument, consider an arbitrary point in the plane, O, as origin. Then take two lines, $\displaystyle l$ and $\displaystyle m$, through O parallel to $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$ respectively (effectively setting up a system of non-perpendicular* coordinate axes). Then consider point C whose displacement from O is represented by $\displaystyle \vec{c}$. From C, draw a line parallel to $\displaystyle l$, meeting $\displaystyle m$ at P. Then $\displaystyle \vec{PC} = s\vec{a}$ for some scalar $\displaystyle s$, and $\displaystyle \vec{OP} = t\vec{b}$ for some scalar $\displaystyle t$, and then:
$\displaystyle \vec{OC} = \vec{OP} + \vec{PC}$
$\displaystyle \Rightarrow \vec{c} = s\vec{a} + t\vec{b}$
For the components argument, let $\displaystyle \vec{a} = a_1\vec{i} + a_2\vec{j}$, and $\displaystyle \vec{b}$ and $\displaystyle \vec{c}$ similarly. Then consider separate components and show that equations
$\displaystyle c_1 = sa_1 + tb_1$
$\displaystyle c_2 = sa_2 + tb_2$
have a solution for some $\displaystyle s$ and $\displaystyle t$.
Grandad
* PS. Not necessarily perpendicular is what I mean. Instead of being covered by a grid of unit squares, as in Cartesian geometry, the plane is spanned by parallelograms with sides $\displaystyle \vec{a}$ and $\displaystyle \vec{b}$.