Dot Product

#Math

Topics

• Law of Cosines

Properties

$\displaystyle \vec{a}\cdot \vec{b}\equiv ab\cos \theta$

• $\displaystyle \theta$ is the angle between the vectors

$\displaystyle \vec{a}\cdot(\vec{b}+\vec{c})=\vec{a}\cdot \vec{b}+\vec{a}\cdot \vec{c}$

• Dot products are distributive

Linear Algebra

$\displaystyle \mathbf{A}\cdot \mathbf{B}=\mathbf{A}^{\intercal}\mathbf{B}$

Quantum Mechanics

$\displaystyle \braket{ \Psi_{m}, \Psi_{n} }=\braket{ \Psi_{m} | \Psi_{n} }=\int_{-\infty}^{\infty} \Psi_{m}^{*}(x)\Psi_{n}(x) , \mathrm{d}x$