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How to calculate a angle between two vectors in 3D

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How to calculate a angle between two vectors in 3D

Paulo Oliveira asked: AI, Data Science, and Statistics , Statistics and Machine Learning Toolbox , Probability Distributi

Learn how to calculate the angle between two 3D vectors using MATLAB! This resource provides a clear explanation & code examples for vector angle calculations.

Expert Answer

Profile picture of John Michell John Michell answered . 2025-07-11 12:04:51

To calculate angles between two segments in different planes (xyxy, yzyz, xzxz), you need to project the segments onto the respective plane and then compute the angle.


Steps to Calculate Angles for Each Plane

  1. Extract Segment Vectors:

    • For segment S1S1: v1=P2−P1\mathbf{v1} = \mathbf{P2} - \mathbf{P1}
    • For segment S2S2: v2=P7−P1\mathbf{v2} = \mathbf{P7} - \mathbf{P1}
  2. Project Vectors onto Each Plane:

    • XY Plane: Set the zz-component to 0: v1xy=[v1x,v1y,0]\mathbf{v1}_{xy} = [v1_x, v1_y, 0], v2xy=[v2x,v2y,0]\mathbf{v2}_{xy} = [v2_x, v2_y, 0]
    • YZ Plane: Set the xx-component to 0: v1yz=[0,v1y,v1z]\mathbf{v1}_{yz} = [0, v1_y, v1_z], v2yz=[0,v2y,v2z]\mathbf{v2}_{yz} = [0, v2_y, v2_z]
    • XZ Plane: Set the yy-component to 0: v1xz=[v1x,0,v1z]\mathbf{v1}_{xz} = [v1_x, 0, v1_z], v2xz=[v2x,0,v2z]\mathbf{v2}_{xz} = [v2_x, 0, v2_z]
  3. Calculate the Angles:

    • Use the formula for the angle between two vectors: angle=arctan?2(?v1×v2?,v1⋅v2)\text{angle} = \arctan2(\|\mathbf{v1} \times \mathbf{v2}\|, \mathbf{v1} \cdot \mathbf{v2})
    • Convert the result to degrees if needed.

MATLAB Code

Here's the MATLAB implementation:

% Define points
points = [
    0, 0, 0;       % Point 1
    70.5, 0, 0;    % Point 2
    0, 137.5, 0;   % Point 7
];

% Extract vectors for segments
P1 = points(1, :);
P2 = points(2, :);
P7 = points(3, :);

v1 = P2 - P1; % Segment 1: P1 -> P2
v2 = P7 - P1; % Segment 2: P1 -> P7

% Project vectors onto XY, YZ, and XZ planes
v1_xy = [v1(1), v1(2), 0];
v2_xy = [v2(1), v2(2), 0];

v1_yz = [0, v1(2), v1(3)];
v2_yz = [0, v2(2), v2(3)];

v1_xz = [v1(1), 0, v1(3)];
v2_xz = [v2(1), 0, v2(3)];

% Calculate angles in each plane
angle_xy = atan2d(norm(cross(v1_xy, v2_xy)), dot(v1_xy, v2_xy));
angle_yz = atan2d(norm(cross(v1_yz, v2_yz)), dot(v1_yz, v2_yz));
angle_xz = atan2d(norm(cross(v1_xz, v2_xz)), dot(v1_xz, v2_xz));

% Display results
fprintf('Angle in XY plane: %.2f degrees\n', angle_xy);
fprintf('Angle in YZ plane: %.2f degrees\n', angle_yz);
fprintf('Angle in XZ plane: %.2f degrees\n', angle_xz);

Explanation of Results

  1. XY Plane:
    • The zz-components of the vectors are ignored, focusing on the projection in the xyxy-plane.
  2. YZ Plane:
    • The xx-components are ignored.
  3. XZ Plane:
    • The yy-components are ignored.

This will give you the angles between the segments in each of the specified planes. Let me know if you have further questions!


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