This can be viewed as another way of computing the perpendicular distance from an axis to a point, because the matrix formed by the outer product [ "'R " " " R "'] yields the identify
32.
Because the unit basis vectors are orthogonal to each other, the geometric product reduces to the antisymmetric outer product \ scriptstyle \ hat x and \ scriptstyle \ hat y can be swapped freely at the cost of a factor of " 1.
33.
First generate the bivector " B " that is associated with the plane spanned by " a " and " b ", using the outer product from Geometric algebra ( in fact all this is done with Geometric algebra)
34.
Anything in particular about the eigenvectors or eigenvalues of M ? ( No, the function f is not a multiplicative function, which would have made M into a outer product of two vectors . ) talk ) 23 : 25, 15 January 2010 ( UTC)
35.
There is no difference between the forward-time and reversed-time states for the outer product contraction, so here they share the same diagram, represented as one line without direction, again labelled by " j " only and not " m ":
36.
On an inner product space, or more generally a vector space with a nondegenerate form ( so an isomorphism ) vectors can be sent to covectors ( in coordinates, via transpose ), so one can take the inner product and outer product of two vectors, not simply of a vector and a covector.
37.
In the case when the vectors are in the plane z = 0, this degenerates to the unit vector ( 0, 0, 1 ) times the outer product x _ 0 y _ 1-x _ 1 y _ 0; so that outer product is twice the signed area of the triangle.
38.
In the case when the vectors are in the plane z = 0, this degenerates to the unit vector ( 0, 0, 1 ) times the outer product x _ 0 y _ 1-x _ 1 y _ 0; so that outer product is twice the signed area of the triangle.
39.
No indices implies it is a scalar, one implies that it is a vector, etc . Furthermore, any number of new scalars, vectors, etc . can be made by contracting or creating an outer product of any kinds of tensors together, but many of these may not have any real physical meaning.
40.
The energy associated with an electric quadrupole moment in an electric field depends not on the field strength, but on the electric field gradient, confusingly labelled \ scriptstyle { \ underline { \ underline { q } } }, another rank-2 tensor given by the outer product of the del operator with the electric field vector:
