Three points and in three-dimensional Euclidean space have their respective coordinates and What is the measure of
Consider the regular octagon centered at the origin as shown above. Eight unit vectors are drawn from the center of the octagon to each of its vertices. For each pair of distinct unit vectors, the dot product is computed. What is the sum of all of these dot products?
If a line makes angles , , and with the four body diagonals of a cube and the value of
can be expressed as , where and are coprime integers, find the value of .
Clarification: Body diagonals of a cube are the diagonals which do not lie along any face of the cube.
Let be the list of distinct vectors which have 4 coordinates, whose values are either 0 or 1. What is the maximum possible value of ?
Details and assumptions
represents the dot product of vectors.
Examples of vectors which have 4 coordinates and whose entries are either 0 or 1 are: .
The list is a set of all the 16 distinct vectors which satisfy the condition.
Given two vectors and such that and , what is the positive difference between the largest and smallest possible values of