# Coplanar vectors

by yourmom98
Tags: coplanar, vectors
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,682 Do you mean you don't know how to do 3 equations in 3 unknowns? You want to show that W1 = 2V1 + 3V2 , W2 = V2 + 2V3, and W3 = -V1 - 3V3 are coplanar. Three vectors are coplanar if and only if they are dependent- that is if there exist some non-trivial ($$\alpha$$, $$\beta$$, $$\gamma$$ not all equal to 0) such that the linear combination $$\alpha W_1+ \beta W_2+ \gamma W_3= 0$$ $$\alpha(2V_1+ 3V_2)+ \beta(V_2+ 2V_3)+ \gamma(-V_1- 3V_3)= 0$$ $$(2\alpha-\gamma)V_1+ (3\alpha+ \beta)V_2+ (3\beta- 3\gamma)V_3= 0$$ My first thought was that we would need to know more about V1, V2, and V3 but I don't think we do. If V1, V2, and V3 are themselves dependent, then they are coplanar and any linear combination of them is coplanar. So the only question is if V1, V2, and V3 are independent vectors. If V1, V2, and V3 are independent then in order to have $$(2\alpha-\gamma)V_1+ (3\alpha+ \beta)V_2+ (3\beta- 3\gamma)V_3= 0$$ we must have $$2\alpha- \gamma= 0$$ $$3\alpha+ \beta= 0$$ $$3\beta- 3\gamma= 0$$ The three vectors W1, W2, and W3 are coplanar if and only if there exist solutions to those equations other than $$\alpha= \beta= \gamma= 0$$