The vector cross product (or simply vector product, as
opposed to the scalar product (which is also called the
dot product, or inner product)) is commonly used in
vector calculus--a basic mathematical toolset used in
acoustics , electromagnetism , quantum
mechanics, and more. It can be defined symbolically in the form of
a matrix determinant:B.19
where denote the unit vectors in . The cross-product is a vector in 3D that is orthogonal to the plane spanned by and , and is oriented positively according to the right-hand rule.B.20
The second and third lines of Eq.(B.15) make it clear that . This is one example of a host of identities that one learns in vector calculus and its applications.
where denotes the identity matrix in , denotes the orthogonal-projection matrix onto , denotes the projection matrix onto the orthogonal complement of , denotes the component of orthogonal to , and we used the fact that orthogonal projection matrices are idempotent (i.e., ) and symmetric (when real, as we have here) when we replaced by above. Finally, note that the length of is , where is the angle between the 1D subspaces spanned by and in the plane including both vectors. Thus,
The direction of the cross-product vector is then taken to be orthogonal to both and according to the right-hand rule. This orthogonality can be checked by verifying that . The right-hand-rule parity can be checked by rotating the space so that and in which case . Thus, the cross product points ``up'' relative to the plane for and ``down'' for .
To see this, let's first check its direction and then its magnitude. By the right-hand rule, points up out of the page in Fig.B.4. Crossing that with , again by the right-hand rule, produces a tangential velocity vector pointing as shown in the figure. So, the direction is correct. Now, the magnitude: Since and are mutually orthogonal, the angle between them is , so that, by Eq.(B.16),
Angular Velocity Vector