Right-hand rule for a general vector product a⨯b:
% Author: Izaak Neutelings (July 2018)
\documentclass[border=3pt,tikz]{standalone}
\usepackage{tikz}
\usepackage{physics}
\tikzset{>=latex} % for LaTeX arrow head
\usetikzlibrary{angles,quotes} % for pic (angle labels)
\usepackage{xcolor}
\colorlet{pinkskin}{pink!25}
\colorlet{brownskin}{pink!5!brown!45}
\colorlet{myred}{red!90!black}
\colorlet{myblue}{blue!90!black}
\colorlet{mypurple}{blue!50!red!80!black!80}
\colorlet{Bcol}{violet!90}
\colorlet{BFcol}{red!60!black}
\colorlet{veccol}{green!45!black}
\colorlet{Icol}{blue!70!black}
\colorlet{mucol}{red!90!black}
\tikzstyle{BField}=[->,line width=2,Bcol]
\tikzstyle{current}=[->,Icol] %thick,
\tikzstyle{force}=[->,line width=2,BFcol]
\tikzstyle{vector}=[->,line width=2,veccol]
\tikzstyle{thick vector}=[->,line width=2,veccol]
\tikzstyle{mu vector}=[->,line width=2,mucol]
\tikzstyle{velocity}=[->,line width=2,veccol]
\tikzstyle{charge+}=[very thin,draw=black,
top color=red!50,bottom color=red!90!black,shading angle=20,circle,inner sep=0.5]
\begin{document}
\Large
\begin{tikzpicture}
\coordinate (O) at (1.2,0.3); % ORIGIN
\coordinate (WT) at ( 2.9,-1.1); % WRIST TOP
\coordinate (T1) at ( 2.3, 0.7); % THUMB
\coordinate (T2) at ( 1.75, 2.3);
\coordinate (T3) at ( 2.0, 3.1);
\coordinate (T4) at (1.38, 3.15);
\coordinate (T5) at ( 0.9, 2.3);
\coordinate (T6) at ( 0.85, 1.2);
\coordinate (T7) at ( 0.85, 0.2);
\coordinate (I1) at (-1.0, 2.4); % INDEX
\coordinate (I2) at (-2.9, 3.45);
\coordinate (I3) at (-3.3, 2.9);
\coordinate (I4) at (-1.5, 1.8);
\coordinate (I5) at (-0.9, 1.1);
\coordinate (I6) at (-0.9, 0.5);
\coordinate (M1) at (-2.2, 1.25); % MIDDLE
\coordinate (M2) at (-3.9, 1.4);
\coordinate (M3) at (-4.0, 0.8);
\coordinate (M4) at (-2.3, 0.5);
\coordinate (M5) at (-1.1, 0.25);
\coordinate (R1) at (-1.9,-0.1); % RING
\coordinate (R2) at (-1.8,-0.7);
\coordinate (R3) at (-0.3,-1.5);
\coordinate (R4) at ( 0.1,-1.7);
\coordinate (R5) at ( 0.1,-1.0);
\coordinate (R6) at (-0.5,-0.7);
\coordinate (R7) at (-1.2,-0.3);
\coordinate (P1) at (-1.9,-1.3); % PINKY
\coordinate (P2) at (-0.8,-1.9);
\coordinate (P3) at (-0.2,-2.1);
\coordinate (P4) at (-0.05,-1.65);
\coordinate (W1) at ( 0.4,-2.9); % WRIST BOTTOM
\coordinate (W2) at ( 1.6,-3.5);
% HAND
\fill[brownskin]
(WT) -- (T6) -- (I5) -- (M5) -- (R2) -- (P2) -- (W2) to[out=25,in=-90] cycle;
\draw[fill=brownskin]
(WT) to[out=120,in=-60] % THUMB
(T1) to[out=120,in=-90]
(T2) to[out=80,in=-110]
(T3) to[out=80,in=50,looseness=1.5] % tip
(T4) to[out=-130,in=80]
(T5) to[out=-100,in=70]
(T6) to[out=-100,in=100]
(T7)
(T6) to[out=150,in=-30] % INDEX
(I1) to[out=150,in=-30]
(I2) to[out=150,in=145,looseness=1.7] % tip
(I3) to[out=-30,in=150]
(I4) to[out=-30,in=105]
(I5) to[out=-75,in=100]
(I6)
(I5) -- % MIDDLE
(M1) --
(M2) to[out=170,in=180,looseness=1.5] % tip
(M3) to[out=-5,in=175]
(M4) to[out=-5,in=165] % bottom knuckle
(M5)
(M5) to[out=-160,in=50] % RING
(R1) to[out=-130,in=140,looseness=1.2]
(R2) to[out=-30,in=160]
(R3) --
(R4) to[out=-20,in=-20,looseness=1.5] % tip
(R5) --
(R6) to[out=140,in=8,looseness=0.9]
(R7)
(R2) to[out=-160,in=155] % PINKY
(P1) to[out=-35,in=150]
(P2) to[out=-30,in=160]
(P3) to[out=-20,in=-30,looseness=1.5] % tip
%(P4) --
(R4)
(P2) to[out=-50,in=140] % WRIST
(W1) to[out=-40,in=160]
(W2);
% FOLDS
\draw[very thin] (T5)++(-80:0.3) to[out=40,in=180]++ (25:0.45);
\draw[very thin] (I1)++(180:0.2) to[out=-160,in=100]++ (-130:0.6);
\draw[very thin] (I1)++(155:1.3) to[out=-160,in=90]++ (-135:0.55);
\draw[very thin] (M4)++(140:0.1) to[out=110,in=-140]++ (80:0.6);
\draw[very thin] (M3)++(-5:0.6) to[out=100,in=-130]++ (80:0.5);
\draw[very thin] (M5)++(-140:0.1) to[out=-20,in=90]++ (-54:0.8); % RING
\draw[very thin] (R6) to[out=160,in=10]++ (180:0.2);
\draw[very thin] (R3)++(155:0.5) to[out=120,in=-100]++ (100:0.2);
\draw[very thin] (P2)++(140:0.1) to[out=95,in=-110]++ (80:0.4);
%\draw[very thin] (P1)++( 10:0.04) to[out=95,in=-130]++ (70:0.4);
\draw[very thin] (I5)++(-40:0.45) to[out=-70,in=90]++ (-70:1.7); % PALM
\draw[very thin] (P3)++(-155:0.05) to[out=-120,in=40]++ (-130:0.2); % PALM
\draw[very thin] (W2)++(80:1.3) to[out=-180,in=-50]++ (160:1.2); % PALM
% VECTORS
\draw[thick vector,mypurple]
(O) --++ (85:3.4)
node[above,scale=1.5] {${\color{myblue}\vb{a}} \times {\color{myred}\vb{b}}$};
\draw[thick vector,myblue]
(O) --++ (145:3.7) coordinate (A)
node[above=2,left=-7,scale=1.5] {$\vb{a}$};
\draw[thick vector,myred]
(O) --++ (172:3.7) coordinate (B)
node[above=4,left=-5,scale=1.5] {$\vb{b}$};
\draw pic[->,"\huge$\theta$",draw=black,thick,angle radius=35,angle eccentricity=1.24] {angle = A--O--B};
\end{tikzpicture}
\end{document}