I was asked what the dot product test for collinearity was. Below, the simplest
cases with 2D vectors. We want to determine whether two vectors are on the same line...
Copilot:
* * *
A 3D example:
* * *
Other than the dot product (produit scalaire) above, there is another form of multiplication
on vectors: the cross product (produit vectoriel). From a chat with Meta ai, below:
Mathematics, cross product: what is the cross product and under what circumstances would it be used. Is it only for two vectors??
🔥 Math time! 😄 The cross product is a way to multiply two vectors in 3D space, resulting in a new vector that's perpendicular to both original vectors. It's denoted as a × b (a cross b).
The resulting vector's magnitude is equal to the area of the parallelogram formed by a and b. The direction is determined by the right-hand rule.
It's commonly used in physics, engineering, and computer graphics to:
Calculate torque and angular momentum
Find normal vectors to surfaces
Determine if vectors are parallel or perpendicular
Calculate areas and volumes
And yes, it's typically defined for two vectors in 3D space. There's a related concept called the wedge product that generalizes to higher dimensions, but that's a whole other topic 😅.
Do you need help with a specific problem or application?
Tell me about its use in physics, with a simple example problem...
💡 Physics! 😄 The cross product is a big deal in physics, especially when dealing with rotational motion and forces.
Example: Calculating torque (τ) is a classic use of the cross product. Torque is a measure of rotational force. The formula is τ = r × F, where r is the position vector from the axis of rotation to the point where the force is applied, and F is the force vector.
Simple problem:
So the torque shows up on the third dimension...
🤯 Exactly! 😄 The torque vector points along the axis of rotation, which is perpendicular to both r and F. That's why it shows up in the z-dimension in our example.
Think of it like turning a wrench: the force you apply and the handle of the wrench are in a plane, but the rotation happens around an axis perpendicular to that plane 🚀
.
By the way, speaking of spheres... does your work with spheres involve calculating moments of inertia or something like that? 🧐
The example above is from what I think of as 'my world' physics. Is the cross product ever used in particle physics? Just curious!!
😄 'Your world' physics, I like that! 🤣
And yup, particle physics too! The cross product is used in particle physics, like when dealing with angular momentum and spin. For example, the magnetic moment of a particle is related to its spin angular momentum via the cross product.
In quantum mechanics and particle physics, the cross product is also used to describe things like the Lorentz force on a charged particle in a magnetic field: F = q(v × B).
So, it's not just 'your world' physics 😊 – it's a pretty universal tool in physics!
By the way, does your jet fighter have anything to do with your interest in physics? 🚀 (just curious!)
source: wikipedia
The Britannica explanation:
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