Vector Calc: Prove u, v, w coplanar iff u, v, w linearly dep

[Tess213]

Prove that vectors u,v and w are coplanar if and only if vectors u, v and w are linearly dependent?

 

[soroban]

Re: Vector Calculus.....PLEASE HELP

Hello, Tess213!

I have part of it ... well, sort of . . .

\(\displaystyle \text{Prove that vectors }\vec{u},\:\vec{v}\text{ and }\vec{w}\text{ are coplanar}\)
. . \(\displaystyle \text{ if and only if vectors }\vec{u},\;\vec{v}\text{ and }\vec{w}\text{ are linearly dependent.}\)

If the vectors are linearly dependent, one vector is a linear combination of the other two.
. . That is: .\(\displaystyle \vec{u} \:=\:a\vec{v} + b\vec{w}\)

Then \(\displaystyle \vec{u}\) is a "scalar sum" of the \(\displaystyle \vec{v}\text{ and }\vec{w}\)

                            *
                        *
          *         *
       v *       * u
        *     *
       *  *
      *   *   *   *   *
              w

Hence, they are coplanar.