Coplanar and Linear dependency.

[jrotmensen]

Homework Statement

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)

 

[HallsofIvy]

Homework Statement

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)

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.