Formulas and definition of inner product (vector)

Let $\vec{a}$, $\vec{b}$ be vectors in the plane.

\[ \vec{a} = (a_1,\ a_2) \\~\\ \vec{b} = (b_1,\ b_2) \]

The inner product of two vectors is defined as follows.

\[ \vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 \]

Let $\theta$ be the angle between those vectors.

\[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{a}|\textnormal{cos}\theta \]

The angle between two vectors

\[ \textnormal{cos}\theta = \frac{ \vec{a} \cdot \vec{b} }{ |\vec{a}||\vec{a}| } \]

Vector formula

Formula