Hello qbkr21For the geometric argument, consider an arbitrary point in the plane, O, as origin. Then take two lines, and , through O parallel to and respectively (effectively setting up a system of non-perpendicular* coordinate axes). Then consider point C whose displacement from O is represented by . From C, draw a line parallel to , meeting at P. Then for some scalar , and for some scalar , and then:

For the components argument, let , and and similarly. Then consider separate components and show that equations

have a solution for some and .

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 and .