# Coplanar and Linear dependency.

1. Dec 26, 2009

### jrotmensen

1. The problem statement, all variables and given/known data
Prove that vectors u, v, w are coplanar if and only if vectors u, v and w are linearly dependent.

$$\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}$$ (Coplanar Vector Property)
$$\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\overline{0}$$ (linearly dependent vector property)

Last edited: Dec 26, 2009
2. Dec 26, 2009

### HallsofIvy

Staff Emeritus
State the entire properties! What you have are equations, not properties.

"Three vectors, $v_1$, $v_2$, and $v_3$ are coplanar if and only if
$$\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}$$
or
$$\overline{v}_{1}=\alpha\overline{v}_{2}+\beta\overline{v}_{3}$$
or
$$\overline{v}_{2}=\alpha\overline{v}_{1}+\beta\overline{v}_{3}$$
for some numbers $\alpha$ and $\beta$"

"Three vectors, $v_1$, $v_2$, and $v_3$ are dependent if $$\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\overline{0}$$
with not all of $\alpha$, $\beta$, $\gamma$ equal to 0."

Suppose $\vec{v_1}$, $\vec{v_2}$, and $\vec{v_3}$ are planar. Subtract the right side of that equation from both sides.

Suppose $\vec{v_1}$, $\vec{v_2}$, and $\vec{v_3}$ are dependent. Solve that equation for one of the vectors.

Last edited: Dec 26, 2009