Finding Limit of Angle Theta in Unit Cube in R^n

[mrs.malfoy]

Let \theta be the angle between the diagonal of the unit cube in R^{n} and one of its axes.

Find

lim \theta (n)
_{n\rightarrow\infty}

 

[Rasalhague]

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

\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

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