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What's the difference between a wedge product and a cross product?
The wedge product and cross product are both operations used in vector calculus, but they have different geometric interpretations and results. The wedge product is used to find the area or volume spanned by two vectors, while the cross product is used to find a vector perpendicular to two given vectors.
The wedge product of two vectors, u and v, is calculated by taking the magnitude of the cross product of the two vectors and multiplying it by the sine of the angle between them. This can also be written as ||u x v|| * sin(theta).
No, the wedge product is not commutative. In other words, u ∧ v ≠ v ∧ u. This means that the order in which the vectors are multiplied matters and will affect the result.
The cross product can be seen as a special case of the wedge product, where the vectors are in three-dimensional space and the result is a vector perpendicular to both of the original vectors. In other words, the cross product is a three-dimensional version of the wedge product.
The wedge product and cross product have many applications in physics, engineering, and computer graphics. They are used to calculate torque and angular momentum in physics, determine the direction of magnetic fields, and in 3D graphics to create realistic lighting and shading effects.