Cross Product in a sentence

Rotating systems The mathematical description of rotational forces such as torque and angular momentum make use of the cross product The cross product in relation to the outer product.

The cross product differs from the dot product primarily in that the result of the cross product of two vectors is a vector.

Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product.

An illustration of the cross product The cross product a × b is defined so that a, b, and a × b also becomes a right-handed system (but note that a and b are not necessarily orthogonal ).

The cross product can be written as : For arbitrary choices of spatial orientation (that is, allowing for left-handed as well as right-handed coordinate systems) the cross product of two vectors is a pseudovector instead of a vector (see below).

Grassmann's 1840 work Theorie der Ebbe und Flut (Theory of the Ebb and Flow) was the first system of spatial analysis similar to today's system and had ideas corresponding to the cross product, scalar product and vector differentiation.

In classical mechanics, the angular momentum of a particle can be reinterpreted as a plane element: : in which the exterior product ∧ replaces the cross product × (these products have similar characteristics but are nonequivalent).

The cross product of two vectors in 3 dimensions with positive-definite quadratic form is closely related to their outer product.

Any force directed parallel to the particle's position vector does not produce a torque. citation It follows from the properties of the cross product that the torque vector is perpendicular to both the position and force vectors.

As the cross product of a finite number of compact convex sets, is also compact and convex.

Commutator and cross product The commutator of two octonions x and y is given by : This is antisymmetric and imaginary.

For example, : Relation to dot and cross product Pauli vectors elegantly map these commutation and anticommutation relations to corresponding vector products.

In three dimensions, it is further possible to define a cross product which supplies an algebraic characterization of the area and orientation in space of the parallelogram defined by two vectors (used as sides of the parallelogram).

It should be clear that the multiplication rule used for the orientational symbols is not the same as that for the cross product of two vectors.

Let vectors AC and BD form the diagonals from A to C and from B to D. The area of the quadrilateral is then : which is half the magnitude of the cross product of vectors AC and BD.

Now using the definition of force (whether or not mass is constant) and the definition of velocity : The cross product of momentum with its associated velocity is zero because velocity and momentum are parallel, so the second term vanishes.

Pseudovectors occur most frequently as the cross product of two ordinary vectors.

The force F acting on a particle of electric charge q with instantaneous velocity v, due to an external electric field E and magnetic field B, is given by: : where × is the vector cross product.

The vector cross product can be evaluated as the determinant of a matrix : : where the vectors i, j, k are unit vectors in the x, y and z directions.

Torque is the cross product of a distance vector and a force vector.