Add vector
Multiply by scalar
x-axis: \(\langle x, -y \rangle\) y-axis: \(\langle -x, y \rangle\) diagonal: \(\langle y, x \rangle\)
\[ \langle x \cos{\alpha} - y \sin{\alpha}, x \sin{\alpha} + y \cos{\alpha} \rangle \]
\[ \langle x, y \rangle = \langle \cos{\theta}, \sin{\theta} \rangle \]
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos{\theta} \] or \[ \vec{a} \cdot \vec{b} = a_{x} b_{x} + a_{y} b_{y} \] more generally \[ \vec{v} \cdot \vec{w} = \sum\limits_{i=1}^{n} v_{i} w_{i} = ||\vec{v}||\, ||\vec{w}|| \cos{\theta} , \quad \vec{v}, \vec{w} \in \mathbb{R}^{n} \]
A vector space \(V\) is a non-empty set of objects \(\ket{v}\) with addition and scalar multiplication that satisify: - additive associativity - multiplicative associativity - addititive commutativity - existence of \(\ket{0}\) such that \(\ket{v} + \ket{0} = \ket{v}\) and \(\ket{v} + (-1)\ket{v} = \ket{0}\) - multiplicitive identity \(1\ket{v} = \ket{v}\) for all elements in \(V\) - scalar distributivity \(a(\ket{v} + \ket{w}) = a\ket{v} + a\ket{w}\) - vector distributivity \((a + b)\ket{v} = a\ket{v} + b\ket{v}\)