# How to normalize unit vector in ode45?

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Sam H on 23 May 2018
Commented: Torsten on 7 Dec 2018
for example I want to solve a unit vector rotating around an axis. the kinematics would be: i_dot = w cross i, where 'cross' means cross product, ' i' is the unit vector and ' w' is angular velocity vector. Then ode45 is used to solve this equation. However, because of the numerical integration, length of the unit vector will drift away from unity, how can I normalize the vector in every loop? Thanks! Example code:
clear;clc
global w
w = [0;0;1]; % rotates around z axis, with angular velocity 1 rad/s
i0 = [1/2;1/2;1/sqrt(2)]; % initial unit vector
tspan = [0,100]; % time span
[t,i] = ode45(@exampleFun,tspan,i0); % use ode45 to solve for new unit vector
function idot = exampleFun(t,i)
global w
idot = cross(w,i);
end

Torsten on 23 May 2018
function main
w = [0;0;1]; % rotates around z axis, with angular velocity 1 rad/s
i0 = [1/2;1/2;1/sqrt(2)]; % initial unit vector
tspan = [0,100]; % time span
opts = odeset('RelTol',1e-8,'AbsTol',1e-8);
[t,i] = ode45(@(t,i)exampleFun(t,i,w),tspan,i0,opts); % use ode45 to solve for new unit vector
absi=i(:,1).^2+i(:,2).^2+i(:,3).^2;
plot(t,absi)
end
function idot = exampleFun(t,i,w)
idot = cross(w,i);
end
Better ?
Best wishes
Torsten.
##### 2 CommentsShowHide 1 older comment
Torsten on 7 Dec 2018
function idot = exampleFun(t,i,w)
D2i = cross(w,i);
idot = zeros(6,1);
idot(1) = i(2);
idot(2) = D2i(1);
idot(3) = i(4);
idot(4) = D2i(2);
idot(5) = i(6);
idot(6) = D2i(3);
end