Inner Product

A binary operation $⟨\cdot ,\cdot ⟩:V\times V\to F$ satisfying

Axiom	Description
$⟨v,w⟩=\overline{⟨w,v⟩}$	conjugate symmetry
$⟨v,v⟩\ge 0\forall v\in \mathbb{V}$, with $⟨v,v⟩=0\iff v=0$	positive semidefinitiveness
$⟨v,aw+bz⟩=a⟨v,w⟩+b⟨v,z⟩$	Linearity

where

• $v,w,z\in V$ are vectors in the Vector Space $V$
• $a,b,c\in F$ are scalars in the Field $F$.
• $\overline{c}$ is the complex conjugate of $c\in \mathbb{C}$

Angle

Inner products relate to angles in Euclidean Space via the formula $\cos (\theta )=\frac{x\cdot y}{\Vert x\Vert \Vert y\Vert }$. It can be useful to be able to convert between these values, using:

• $\theta =\arccos (x\cdot y)$
• $x\cdot y=\cos (\theta )$

$\theta$	$x\cdot y$	$(x\cdot y{)}^2$
0	1.0000	1.0000
5	0.9962	0.9924
10	0.9848	0.9698
15	0.9659	0.9330
20	0.9397	0.8830
25	0.9063	0.8214
30	0.8660	0.7500
35	0.8192	0.6710
40	0.7660	0.5868
45	0.7071	0.5000
50	0.6428	0.4132
55	0.5736	0.3290
60	0.5000	0.2500
65	0.4226	0.1786
70	0.3420	0.1170
75	0.2588	0.0670
80	0.1736	0.0302
85	0.0872	0.0076
90	0.0000	0.0000