## Problem on finding the angle between vectors Find the angle at the apex of a triangular faces of the pyramid formed by the points $(1, 1, 0)$, $(1,-1, 0)$, $(-1, 1, 0)$, $(-1, -1, 0)$, and $(0,0,1)$.  • ## Solution To find the angle at the apex, we will express two adjacent edges as vectors and compute the angle between those vectors. Recall that As a result, we represent two edges as vectors whose base is the apex of the pyramid:   Let's consider the edges going from the apex $(0,0,1)$ to either $(1,1,0)$ or $(1,-1,0)$. Recall that Hence, these two edges are represented by the vectors $(1,1,0) -(0,0,1) = (1,1,-1)$ and $(1,-1,0) - (0,0,1) = (1,-1,-1).$ We have reduced the problem to finding the angle between the vectors $(1,1,-1)$ and $(1,-1,-1)$. Recall the relationship between dot products and angles: To use this formula, we let $\bfx = (1,1,-1)$ and $\bfy = (1,-1,-1).$ By the dot product angle formula, $$1 = \sqrt{3} \sqrt{3} \cos \theta.$$ We conclude that $\theta = \cos^{-1} \frac{1}{3} \approx 1.2$ radians $\approx 71 ^\circ$.