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Modern Geometry

  1. Dec 14, 2009 #1
    Let [tex]\theta[/tex] be the angle between the diagonal of the unit cube in [tex]R^{n}[/tex] and one of its axes.

    Find

    lim [tex]\theta[/tex] (n)
    [tex]_{n\rightarrow\infty}[/tex]
     
  2. jcsd
  3. Dec 14, 2009 #2
    You can represent an edge as a unit vector along the x-axis, (1,0,0), and the diagonal as the vector (1,1,1). Consider the definition of the dot product

    [tex]\mathbf{a} \cdot \mathbf{b}= a_{1}b_{1}+a_{2}b_{2}+...+a_{n}b_{n} = \left \| a \right \| \left \| b \right \| cos \; \theta[/tex]

    Since we know a = (1,0,0,...) and b = (1,1,1,...), we can solve for [itex]\theta[/itex] in terms of n. To find the limit as n approaches infinity, just try it out with very big numbers.
     
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