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Homework Help: Vector equations

  1. Oct 3, 2005 #1
    Let [itex]a_1[/itex] = a column vector with 1, 4, -2; [itex]a_2[/itex] = a column vector with -2, -3, 7; and [itex]b[/itex] = a column vector with entries 4, 1, h.
    (I hope this is an adequate description. I forgot how to write pretty matrices in tex ^_^;)

    For what values of h is [itex]b[/itex] in the plane spanned by [itex]a_1[/itex] and [itex]a_2[/itex]?

    I turned this into an augmented matrix but had trouble reducing it to RREF.
     
  2. jcsd
  3. Oct 4, 2005 #2

    CarlB

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    Homework Helper

    I had the best success in vectors when I put everything in terms of dot products and cross products. In this problem, one can take [itex]C = a_1 \times a_2[/itex] as defining the plane spanned by [itex]a_1[/itex] and [itex]a_2[/itex]. Then h is in the plane if [itex]h \cdot C[/itex] is zero. That is, if C is perpendicular to h.

    Click on this example to be reminded how to format matrices in LaTex with various boundary definitions &c:

    [tex]\left( \left[ \begin{array}{ccc}
    0 & 1 & 2 \\
    3 & 4 & 5 \end{array} \right| \right)[/tex]

    Carl
     
  4. Oct 4, 2005 #3

    TD

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    Put them in a matrix and compute the determinant. If det is 0, then the vector are coplanar and thus, every vector is in the plane span by the other two.

    [tex]\left| {\begin{array}{*{20}c}
    1 & { - 2} & 4 \\
    4 & { - 3} & 1 \\
    { - 2} & 7 & h \\

    \end{array} } \right| = 0 \Leftrightarrow h = - 17[/tex]
     
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