How to calculate a angle between two vectors in 3D

Paulo Oliveira - 2024-01-11T14:02:08+00:00

Question: How to calculate a angle between two vectors in 3D

Hi, I have a question, I have a table with 12 reference points   if true % POINT X Y Z 1 0 0 0 2 70.5 0 0 3 141 0 0 4 141 0 141.5 5 70.5 0 141.5 6 0 0 141.5 7 0 137.5 0 8 70.5 137.5 0 9 141 140 0 10 141 141.5 141.5 11 70.5 139 141.5 12 0 141.5 141.5 end The segment 1 is defined by the point 1 and point 2 and the segment 2 is defined by the point 1 and point 7. I am able to calculate the angle with the following rotine,     if true % angle_x1_x13 = atan2(norm(cross(v1,v2)),dot(v1,v2)); anglout = radtodeg(angle_x1_x13); end The result is ~90º and this result is correct if I think in xy plan, but I need to calculate the angle to yz plan and xz plan. Anyone help me?  

Related Questions

Expert Answer

John Michell answered . 2025-11-20

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!

Not satisfied with the answer ?? ASK NOW