Search results
Results From The WOW.Com Content Network
In geometry and algebra, the triple product is a product of three 3-dimensional vectors, usually Euclidean vectors.The name "triple product" is used for two different products, the scalar-valued scalar triple product and, less often, the vector-valued vector triple product.
To elucidate the connection with the triple product rule, consider the point p 1 at time t and its corresponding point (with the same height) p̄ 1 at t+Δt. Define p 2 as the point at time t whose x-coordinate matches that of p̄ 1 , and define p̄ 2 to be the corresponding point of p 2 as shown in the figure on the right.
The following are important identities in vector algebra.Identities that only involve the magnitude of a vector ‖ ‖ and the dot product (scalar product) of two vectors A·B, apply to vectors in any dimension, while identities that use the cross product (vector product) A×B only apply in three dimensions, since the cross product is only defined there.
The dotted vector, in this case B, is differentiated, while the (undotted) A is held constant. The utility of the Feynman subscript notation lies in its use in the derivation of vector and tensor derivative identities, as in the following example which uses the algebraic identity C⋅(A×B) = (C×A)⋅B:
Vector algebra relations — regarding operations on individual vectors such as dot product, cross product, etc. Vector calculus identities — regarding operations on vector fields such as divergence, gradient, curl, etc.
The torque or curl is then a normal vector field in this 3rd dimension. By contrast, geometric algebra in 2 dimensions defines these as a pseudoscalar field (a bivector), without requiring a 3rd dimension. Similarly, the scalar triple product is ad hoc, and can instead be expressed uniformly using the exterior product and the geometric product.
Basis Decomposition of a 2-vector. For vectors in R 3, the exterior algebra is closely related to the cross product and triple product.Using the standard basis {e 1, e 2, e 3}, the exterior product of a pair of vectors
For the case of two separate closed wires, the law can be rewritten in the following equivalent way by expanding the vector triple product and applying Stokes' theorem: [7] = ( ) ^ | |. In this form, it is immediately obvious that the force on wire 1 due to wire 2 is equal and opposite the force on wire 2 due to wire 1, in accordance with ...