The Power of Cross Products: A Visual Guide to Precessing Vectors

The precession of a vector is a direct consequence of a cross product in its differential equation, which ensures that the vector's magnitude and its angle relative to the precession axis remain constant. This stability arises because the change in the vector ( $d L / d t$ ) is always perpendicular to the vector itself and the constant vector it's precessing around.

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✍️Mathematical Proof

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The Magnitude of $L$ is Constant

The derivation shows that the time derivative of the squared magnitude, $\frac{d}{d t}(L \cdot L)$, is zero. This is because the cross product $v \times L$ is always perpendicular to $L$. When we substitute the differential equation $\frac{d L}{d t}=v \times L$ into the derivative, we get a dot product of two orthogonal vectors, which is always zero. This proves that the length of the angular momentum vector does not change over time, even as its direction changes.

The Inner Product of $L$ and $v$ is Constant

Similarly, the analysis shows that the time derivative of the inner product $\frac{d}{d t}(L \cdot v)$ is also zero. This is a direct consequence of the fact that $v$ is a constant vector and the cross product $v \times L$ is always perpendicular to $v$. The constancy of this inner product, along with the constant magnitude of both $L$ and $v$, means that the angle between the two vectors, $L$ and $v$, is always constant.

Physical Interpretation

These two mathematical results perfectly describe the physical motion of precession. The angular momentum vector $L$ rotates around the constant vector $v$ while maintaining a fixed length and a fixed angle relative to $v$, creating a conical motion. The analysis proves that this specific form of rotation is an inherent property of the given differential equation.

🎬Demonstration

the precession of a vector is a direct consequence of a cross product in its differential equation. The simulation visually demonstrates how the change in $L$ (i.e., $d L / d t$ ) is always perpendicular to both $v$ and $L$. Because the change is always perpendicular to $L$ itself, the length of $L$ doesn't change, only its direction. The interactive sliders let you set the initial conditions, making it clear that while the specific magnitude and inner product of $L$ depend on its starting state, those values are then held constant throughout the precession, precisely as the mathematical proof predicts.

a 3D simulation of vector precession