Geometry
# Dot Product of Vectors

If a line makes angles $\alpha$ , $\beta$ , $\gamma$ and $\delta$ with the four body diagonals of a cube and the value of

$\cos^2(\alpha) + \cos^2(\beta) + \cos^2(\gamma) + \cos^2(\delta)$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are coprime integers, find the value of $p + q$.

**Clarification:** Body diagonals of a cube are the diagonals which do not lie along any face of the cube.

Let $a_1, \ldots, a_{16}$ be the list of $2^4 = 16$ distinct vectors which have 4 coordinates, whose values are either 0 or 1. What is the maximum possible value of $a_1 \cdot a_2 + a_3 \cdot a_4 + \cdots + a_{15} \cdot a_{16}$?

**Details and assumptions**

$u \cdot v$ represents the dot product of vectors.

Examples of **vectors which have 4 coordinates and whose entries are either 0 or 1** are: $(0, 0, 0, 0), (1, 1, 1, 1), (0, 1, 0, 1), (1, 0, 0, 0)$.

The list is a set of all the 16 distinct vectors which satisfy the condition.