Modern Geometry (1 Viewer)

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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]
 
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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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