Consider two vectors and in . Show that a may be written as the sum of two vectors: one parallel (or anti-parallel) to and the other perpendicular to . By setting the former equal to , where is a unit vector along , show that
Explain why this is a sensible definition of the angle between and .
Consider the vertices of a cube of side 2 in , centered on the origin. Each vertex is joined by a straight line through the origin to another vertex: the lines are the diagonals of the cube. Show that no two diagonals can be perpendicular if is odd.
For , what is the greatest number of mutually perpendicular diagonals? List all the possible angles between the diagonals.