Questions about pararell planes in R3.

Hi!

I got two parallel planes in R3 given by the equations 3x + 4y -5z = 0 & 3x + 4y - 5z = 20. I have plotted them with

z1 = @(x,y) 0.6*x + 0.8*y;
z2 = @(x,y) 0.6x + 0.8*y -4;
fsurf(z1);
hold on
fsurf(z2);

I would like to find the normal vector by using matlab, I know its [3 4 -5] from the equations but I'd like to know how I cand find it using matlab.

Then I would like to plot the normal vectors between the planes, and measure the distance between the planes. So my questions are;

How do I find a normal vector to a plane in Matlab?

How do I plot it between the two planes?

And how do I measure the distance?

Best regards

[Aitor]

Did you ever get an answer?

If you did, can you tell me how to do it please.

Answer 1

[Abhinaya Kennedy]

Finding the Normal Vector:

Since you've already identified the normal vector as ([3, 4, -5]), you can directly use this in MATLAB. However, if you wish to extract it programmatically from the equation, you can define it as follows:

% Coefficients of the plane equations
a = 3;
b = 4;
c = -5;

% Normal vector
normal_vector = [a, b, c];

Plotting the Normal Vector:

To plot the normal vector between the two parallel planes, you can use a point on one of the planes and draw the vector.

% Define the planes
z1 = @(x, y) 0.6*x + 0.8*y;
z2 = @(x, y) 0.6*x + 0.8*y - 4;

% Plot the planes
fsurf(z1, [-10, 10, -10, 10]);
hold on;
fsurf(z2, [-10, 10, -10, 10]);

% Choose a point on the first plane (e.g., x=0, y=0)
point_on_plane1 = [0, 0, z1(0, 0)];

% Calculate a point on the second plane using the normal vector
point_on_plane2 = point_on_plane1 + 4 * (normal_vector / norm(normal_vector));

% Plot the normal vector
quiver3(point_on_plane1(1), point_on_plane1(2), point_on_plane1(3), ...
        normal_vector(1), normal_vector(2), normal_vector(3), 1, 'r', 'LineWidth', 2);

% Plot the line between the two points
line([point_on_plane1(1), point_on_plane2(1)], ...
     [point_on_plane1(2), point_on_plane2(2)], ...
     [point_on_plane1(3), point_on_plane2(3)], 'Color', 'k', 'LineWidth', 2);

% Adjust plot settings
xlabel('X');
ylabel('Y');
zlabel('Z');
axis equal;
grid on;
hold off;

Measuring the Distance Between the Planes:

The distance (d) between two parallel planes (ax + by + cz = d_1) and (ax + by + cz = d_2) is given by:

[ d = \frac{|d_2 - d_1|}{\sqrt{a^2 + b^2 + c^2}} ]

% Constants from the plane equations
d1 = 0;
d2 = 20;

% Calculate the distance
distance_between_planes = abs(d2 - d1) / norm(normal_vector);

% Display the distance
disp(['Distance between the planes: ', num2str(distance_between_planes)]);

Distance between the planes: 2.8284