Multiple 3d random walks

Hi everyone I have the next code that creates a 3d random walk but i would like to have in the 3d space some random points(seeds) and start from each one of them a random walk at the same time. Can anyone help me?? My code for the random walk is the following:

lamda = 3; %Mean free path 
numberOfSteps = 100000; % Totlal number of steps 
x(1) = rand; % Initial position (x) 
y(1) = rand; % Initial position (y) 
z(1) = rand; % Initial position (z)
for i = 1:numberOfSteps
    r = -lamda*log(rand());  % Distance Travelled
    theta = pi*rand();       % Arbritary angle in between 0 and Pi
    phi = 2*pi*rand();       % Arbritary angle in between 0 and 2Pi

    dx = r*sin(theta)*cos(phi);   % Step Size (x)
    dy = r*sin(theta)*sin(phi);   % Step Size (y)
    dz = r*cos(theta);            % Step Size (z)

    x(i+1) = x(i) + dx;  % Position at the end of the first step (x)
    y(i+1) = y(i) + dy;  % Position at the end of the second step (y)
    z(i+1) = z(i) + dz;  % Position at the end of the third step (z)
end
plot3(x, y, z, 'k');

Answer 1

[Torsten]

lambda = 3; %Mean free path 
numberOfSteps = 10000; % Total number of steps 
numberOfPaths = 3; % Number of paths
x(1,:) = rand(1,numberOfPaths); % Initial position (x) 
y(1,:) = rand(1,numberOfPaths); % Initial position (y) 
z(1,:) = rand(1,numberOfPaths); % Initial position (z)
for j = 1:numberOfPaths
  for i = 1:numberOfSteps
    r = -lambda*log(rand());  % Distance Travelled
    theta = pi*rand();       % Arbritary angle in between 0 and Pi
    phi = 2*pi*rand();       % Arbritary angle in between 0 and 2Pi
    dx = r*sin(theta)*cos(phi);   % Step Size (x)
    dy = r*sin(theta)*sin(phi);   % Step Size (y)
    dz = r*cos(theta);            % Step Size (z)
    x(i+1,j) = x(i,j) + dx;  % Position at the end of the first step (x)
    y(i+1,j) = y(i,j) + dy;  % Position at the end of the second step (y)
    z(i+1,j) = z(i,j) + dz;  % Position at the end of the third step (z)
  end
end
plot3(x(:,1), y(:,1), z(:,1), 'k');
hold on
plot3(x(:,2), y(:,2), z(:,2), 'r');
plot3(x(:,3), y(:,3), z(:,3), 'g');
hold off

[lena kappa]

Thank you @Torsten so much!!! If you could help me with one more thing please:

If I plot the 2D image with view(2) I get something like this :

but is there a way to get a grayscale like image in the sense that if one point has multiple points beneath it in the 3D space it get darker and darker while if it has fewer or no point below it in the 3D space it is brighter???

[Torsten]

Sorry, I don't know.

Maybe you should open a separate question for this.

[lena kappa]

It's ok thank you very much ☺!