How to calculate a angle between two vectors in 3D

John Michell answered . 2025-03-27 14:26:11

To calculate angles between two segments in different planes ($xy$, $yz$, $xz$), 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 $S1$: $v1=P2-P1$
   • For segment $S2$: $v2=P7-P1$

2. Project Vectors onto Each Plane:

   • XY Plane: Set the $z$-component to 0: v1xy=[v1x,v1y,0]\mathbf{v1}_{xy} = [v1_x, v1_y, 0]v1xy?=[v1x?,v1y?,0], v2xy=[v2x,v2y,0]\mathbf{v2}_{xy} = [v2_x, v2_y, 0]v2xy?=[v2x?,v2y?,0]
   • YZ Plane: Set the $x$-component to 0: v1yz=[0,v1y,v1z]\mathbf{v1}_{yz} = [0, v1_y, v1_z]v1yz?=[0,v1y?,v1z?], v2yz=[0,v2y,v2z]\mathbf{v2}_{yz} = [0, v2_y, v2_z]v2yz?=[0,v2y?,v2z?]
   • XZ Plane: Set the $y$-component to 0: v1xz=[v1x,0,v1z]\mathbf{v1}_{xz} = [v1_x, 0, v1_z]v1xz?=[v1x?,0,v1z?], v2xz=[v2x,0,v2z]\mathbf{v2}_{xz} = [v2_x, 0, v2_z]v2xz?=[v2x?,0,v2z?]

3. Calculate the Angles:

   • Use the formula for the angle between two vectors: $$\text{angle}=\arctan 2(?v1\times v2?,v1\cdot v2)$$
   • 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 $z$-components of the vectors are ignored, focusing on the projection in the $xy$-plane.
2. YZ Plane:
   • The $x$-components are ignored.
3. XZ Plane:
   • The $y$-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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