The problem is the following:
If $\bf a$, $\bf b$ and $\bf c$ are coplanar vectors related by $λ\mathbf{a}+μ\mathbf{b}+ν\mathbf{c}=0$, where the constants are non-zero, show that the condition for the points with position vectors $α\mathbf{a}$, $β\mathbf{b}$ and $γ\mathbf{c}$ to be collinear is:
$$\frac{λ}{α} + \frac{μ}{β} + \frac{ν}{γ} = 0.$$
I have tried proving the expression by using $(α\mathbf{a} -β\mathbf{b}) \cdot (α\mathbf{a} - γ\mathbf{c}) = 0$ . By substituting $\bf c$ from the equation of the plane, I end up with an expression in the constants, $\bf a$ and $\bf b$ only, but it is still far from close to what we are looking for. So how do I show that the expression is valid?
Firstly, those 0 should be vectors
Secondly, consider the two equations the given one of the plane and a second equation involving the given condition of collinear.
this means that the vector (αa -βb) is some multiple of (γc-βb)
(αa -βb)=k(γc-βb) is the expression for this
rearrange for αa+(k-1)βb+(-kγc)=0 equating coefficients gives α=γ μ=(k-1)β ν=-kγ these can be rearranged and sum for the desired result