Show that three vectors a, b, c lie in the same plane through the origin if and only if there are threescalars α, β, γ, not all zero, such that αa + βb + γc = 0. Solution. This is an "if and only if" statement, so we can either work through a sequence of equivalent statements, or show that the implication "works both ways". We proceed the latter way. Without loss of generality, we work with vectors in a 3-dimensional vector space.