What does the dot product give us?

Question

I am working through some questions and I have an equation for the position vector of a projectile, namely r, and it's given that k is the vector directly vertical to the starting point of the projectile. Why is it that when you calculate the dot product of r and k (at time t when k is maximised) this gives you the maximum height of the projectile. i.e. what information is the dot product giving us.

Answer 1

Short answer: This goes back to Galileo Galilei, and his theory that motion can be decomposed into independent horizontal and vertical components. Scalar product is the formal machinery that algebraic geometry uses to decompose this way.