Vectors

In what follows are various notes and equations dealing with vectors.

Magnitude

\(\begin{align} \lvert a \rvert &= \sqrt{\sum_{i}^n a_i^2}\\ &= \sqrt{a_1^2 + a_2^2 + \ldots + a_n^2} \end{align}\)

Normalized Vector

\(\hat{a} = \dfrac{a}{\lvert a \rvert}\)

Distance Between Two Points

\(\begin{align} \lvert b - a \rvert &= \sqrt{\sum_{i=1}^n (b_i - a_i)^2}\\ &= \sqrt{(b_1 - a_1)^2 + (b_2 - a_2)^2 + \ldots + (b_n - a_n)^2} \end{align}\)

Vector Projection

\(\text{proj}_ba = \dfrac{a \cdot b}{\lvert b \rvert^2}b\)