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LINEAR ALGEBRA: Orthogonal projeciton of [] onto the subspace of R3 spanned by [], []

  1. Nov 6, 2006 #1
    Question (5.1, #26 -> Bretscher, O.):

    Find the orthogonal projection of [tex]\left[\begin{array}{c} 49 \\ 49 \\ 49 \end{array}\right][/tex] onto the subspace of [tex]\mathbb{R}^3[/tex] spanned by [tex]\left[\begin{array}{c} 2 \\ 3 \\ 6 \end{array}\right][/tex] and [tex]\left[\begin{array}{c} 3 \\ -6 \\ 2 \end{array}\right][/tex].

    [tex]\overrightarrow{x} = \left[\begin{array}{c} 49 \\ 49 \\ 49 \end{array}\right][/tex]

    My Answer:

    Magnitude -> [tex]\sqrt{(2)^2 + (3)^2 + (6)^2} = 7\,\,=\,\,\sqrt{(3)^2 + (-6)^2 + (2)^2}[/tex]


    [tex]\overrrightarrow{u_1} = \left[\begin{array}{c} \frac{2}{7} \\ \frac{3}{7} \\ \frac{6}{7}\end{array}\right][/tex]

    [tex]\overrrightarrow{u_2} = \left[\begin{array}{c} \frac{3}{7} \\ \frac{-6}{7} \\ \frac{2}{7}\end{array}\right][/tex]

    [tex]proj_v \overrightarrow{x} = \left( \overrightarrow{u_1} \cdot \overrightarrow{x} \right) \overrightarrow{u_1} + \left( \overrightarrow{u_2} \cdot \overrightarrow{x} \right) \overrightarrow{u_2}[/tex]

    [tex]proj_v \overrightarrow{x} = \frac{539}{7} \overrightarrow{u_1} + \frac{7}{7} \overrightarrow{u_2} = 77 \overrightarrow{u_1} + \overrightarrow{u_2}[/tex]

    [tex]proj_v \overrightarrow{x} = \left[\begin{array}{c} \frac{157}{7} \\ \frac{225}{7} \\ \frac{464}{7}\end{array}\right][/tex]

    Does this look correct?

    EDIT: IT is incorrect, I now figured it out. Thanks:smile:
     
    Last edited: Nov 6, 2006
  2. jcsd
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