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# Angular and linear velocity, cross product

This page explains how to use the cross product to convert linear velocity to angular velocity (or the reverse).

## From linear to angular

Let's consider the body C, rotating around the point O. The linear speed of point A is given by $$\vec{V}$$.

The angular velocity around O is given by :

$$\vec{\omega} = \frac {\vec{OA} \times \vec{V}} { \| \vec{OA} \| ^2 }$$

## From angular to linear

Assuming the angular velocity is known, the linear speed at point A is given by:

$$\vec{V} = \vec{\omega} \times \vec{OA}$$

## Example

The following Matlab code shows how to convert linear velocity to angular velocity and vice-versa:

close all;
clear all;
clc;

%% Parameters

% Coordinates of point O
O=[0;0;0];
% Coordinates of point A
A=[1;2;0];
% Coordinates of point B
B=[2;1;0];

% Speed applied at point A
VA=[0;2;0];

%% Compute angular velocity

% Compute vector OA
vOA=A-O;
% Compute angular speed around point A
wO=cross (vOA,VA)/(norm(vOA)*norm(vOA));

%% Compute linear speed at point B

vOB=B-O;
VB=cross(wO,vOB);

%% Draw system

plot3(O(1),O(2),O(3));
hold on;
line ([O(1) A(1)],[O(2) A(2)],[O(3) A(3)]);
line ([O(1) B(1)],[O(2) B(2)],[O(3) B(3)]);
text(O(1)+0.2,O(2)+0.2,O(3)+0.2,'O','FontSize',100);
text(A(1)+0.2,A(2)+0.2,A(3)+0.2,'A','FontSize',100);
text(B(1)+0.2,B(2)+0.2,B(3)+0.2,'B','FontSize',100);

% Draw speed at point A

% Draw speed at point O

% Draw speed at point B
legend ([hFA,hTO,hFB],'Linear speed at point A','Angular velocity a point O','Linear speed at point B');