Why 3D plots show complex relationships in MATLAB - Challenge Your Understanding

Challenge - 5 Problems

❓ Predict Output

intermediate

Output of 3D plot matrix multiplication

What is the output of the following MATLAB code snippet that creates a 3D surface plot of a matrix multiplication result?

MATLAB

A = [1 2; 3 4];
B = [2 0; 1 3];
C = A * B;
surf(C);

A A 3D surface plot showing a matrix with values [[2 6]; [10 12]]

B A 3D surface plot showing a matrix with values [[4 6]; [7 12]]

C A 3D surface plot showing a matrix with values [[4 6]; [10 12]]

D A 3D surface plot showing a matrix with values [[4 5]; [10 12]]

💡 Hint

Recall how matrix multiplication works: multiply rows of A by columns of B.

Matrix multiplication of A and B results in [[4 6]; [10 12]]. The surf function plots these values in 3D, showing height as the matrix values.

🧠 Conceptual

intermediate

Why 3D plots reveal complex relationships

Why do 3D plots help us see complex relationships between variables better than 2D plots?

A Because 3D plots show interactions between three variables simultaneously, revealing patterns hidden in 2D.

B Because 3D plots use colors that 2D plots cannot use.

C Because 3D plots always use more data points than 2D plots.

D Because 3D plots are easier to create than 2D plots.

💡 Hint

Think about how adding a third dimension adds more information.

3D plots display three variables at once, allowing us to see how they relate together in space, which 2D plots cannot show.

🔧 Debug

advanced

Identify the error in 3D plot code

What error does the following MATLAB code produce when trying to plot a 3D surface?

MATLAB

x = 1:5;
y = 1:5;
[X,Y] = meshgrid(x,y);
Z = X + Y;
surf(x,y,Z);

A Error: Dimensions of X, Y, and Z must match for surf.

B Error: Z must be a vector, not a matrix.

C Error: Undefined function 'meshgrid'.

D No error, the plot displays correctly.

💡 Hint

Check the inputs to surf and their sizes.

No error, the plot displays correctly. surf accepts vectors x and y when length(x) == size(Z,2) and length(y) == size(Z,1), generating the grid internally just like meshgrid.

📝 Syntax

advanced

Correct syntax for 3D plot with meshgrid

Which option shows the correct syntax to create a 3D surface plot of Z = sin(X) * cos(Y) using meshgrid?

A x = 0:0.1:1; y = 0:0.1:1; [X,Y] = meshgrid(x,y); Z = sin(X)*cos(Y); surf(X,Y,Z);

B x = 0:0.1:1; y = 0:0.1:1; [X,Y] = meshgrid(x,y); Z = sin(X).*cos(Y); surf(X,Y,Z);

C x = 0:0.1:1; y = 0:0.1:1; [X,Y] = meshgrid(x,y); Z = sin(x).*cos(y); surf(X,Y,Z);

D x = 0:0.1:1; y = 0:0.1:1; Z = sin(x).*cos(y); surf(x,y,Z);

💡 Hint

Remember to use element-wise multiplication .* for matrices.

Option B correctly creates meshgrid matrices X and Y, computes element-wise multiplication with .*, and plots with surf.

🚀 Application

expert

Interpreting complex 3D plot relationships

Given a 3D plot of Z = X.^2 - Y.^2, what shape does the surface represent and what does it tell about the relationship between X, Y, and Z?

A A saddle shape showing that Z increases with X squared but decreases with Y squared, indicating opposing effects.

B A dome shape showing Z increases with both X and Y squared.

C A flat plane showing no relationship between variables.

D A bowl shape showing Z decreases with both X and Y squared.

💡 Hint

Think about how positive and negative squared terms affect the surface shape.

The surface is a hyperbolic paraboloid (saddle shape). Z grows positively with X squared and negatively with Y squared, showing complex opposing relationships.