Dot Product in Coordinates
Big Idea: The dot product is a powerful operation that takes two vectors and produces a single number (a scalar). It connects algebra with geometry by relating vector coordinates to the angle between vectors.
1. What is a Vector?
Before diving into the dot product, let's recall that a vector in 2D is described by two coordinates: $\vec{a} = (a_1, a_2)$. It has both magnitude (length) and direction.
The magnitude (length) of $\vec{a} = (a_1, a_2)$ is:
$$|\vec{a}| = \sqrt{a_1^2 + a_2^2}$$