Algebra and Geometry | Part IA, 2005

Consider two vectors a\mathbf{a} and b\mathbf{b} in Rn\mathbb{R}^{n}. Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to b\mathbf{b} and the other perpendicular to b\mathbf{b}. By setting the former equal to cosθab^\cos \theta|\mathbf{a}| \hat{\mathbf{b}}, where b^\hat{\mathbf{b}} is a unit vector along b\mathbf{b}, show that

cosθ=abab\cos \theta=\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|}

Explain why this is a sensible definition of the angle θ\theta between a\mathbf{a} and b\mathbf{b}.

Consider the 2n2^{n} vertices of a cube of side 2 in Rn\mathbb{R}^{n}, centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the 2n12^{n-1} diagonals of the cube. Show that no two diagonals can be perpendicular if nn is odd.

For n=4n=4, what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.

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