Coplanar and Linear dependency.

  1. 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.


    [tex]\overline{v}_{3}=\alpha\overline{v}_{1}+\beta\overline{v}_{2}[/tex] (Coplanar Vector Property)
    [tex]\alpha\overline{v}_{1}+\beta\overline{v}_{2}+\gamma\overline{v}_{3}=\overline{0}[/tex] (linearly dependent vector property)
     
    Last edited: Dec 26, 2009
  2. jcsd
  3. HallsofIvy

    HallsofIvy 40,784
    Staff Emeritus
    Science Advisor

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

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

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

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

    Suppose [itex]\vec{v_1}[/itex], [itex]\vec{v_2}[/itex], and [itex]\vec{v_3}[/itex] are dependent. Solve that equation for one of the vectors.
     
    Last edited: Dec 26, 2009
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