Founded upon the lectures of J . Willard Gibbs, Ph . D ., LL . D . " The first chapter covers scalar triple product and a quadruple product.
12.
The dot product of two vectors is a scalar but the dot product of a pseudovector and a vector is a pseudoscalar, so the scalar triple product must be pseudoscalar-valued.
13.
Combining the above diagrams for the cross product and the dot product, one can read off the three leftmost diagrams as precisely the three leftmost scalar triple products in the above identity.
14.
(For instance, when, i . e . in three-dimensional space, the 2-form is locally the scalar triple product with . ) The integral of over a hypersurface is the flux of over that hypersurface.
15.
Second, the scalar triple product is zero if and only if the three vectors are linearly dependent, which can be easily proved by considering that in order for the three vectors to not make a volume, they must all lie in the same plane.
16.
In the case of interest, u and v are two of the rows, which you've assumed are linearly independent ( so they span a plane ), a third vector x is in the plane spanned by u and v only if there is a linear relation among x, u, and v . This is the case only when det ( x, u, v ) = 0 ( i . e ., their scalar triple product is zero ).
