Rotating a 3D Coordinate System About The Origin

Rotating an orthogonal 3D coordinate system about its axes is tricky because there are multiple reference points to define a rotation and each rotation could be composed of multiple rotations.

To transform a point from the original system to the rotated system, we need to understand how the rotated axes relate to the original axes. If x₁, x₂, and x₃ represent the three original axes and x₁', x₂', and x₃' represent the rotated axes, we need the following nine angles.
Angles Between;

1. x₁' and x₁ - Angle x₁₁
2. x₁' and x₂ - Angle x₁₂
3. x₁' and x₃ - Angle x₁₃
4. x₂' and x₁ - Angle x₂₁
5. x₂' and x₂ - Angle x₂₂
6. x₂' and x₃ - Angle x₂₃
7. x₃' and x₁ - Angle x₃₁
8. x₃' and x₂ - Angle x₃₂
9. x₃' and x₃ - Angle x₃₃

But the way orthogonality works, all nine of these are not independent, instead, only three are independent and the rest of the six can be derived from the three.

Use the sliders present below to switch around the values and see how they interact with each other. Use the checkbox to fix a relation, a maximum of three can be fixed at a time. You can only change the slider whose value is fixed.

Rotational transformation preserves the vector length.

Original Vector:$\sqrt{(0)^2 + (0)^2 + (0)^2} = 0.00$

New Vector:$\sqrt{(0)^2 + (0)^2 + (0)^2} = 0.00$