Derive an expression for the triple scalar product in terms of the determinant of the matrix whose rows are given by the components of the three vectors .
Use the geometrical interpretation of the cross product to show that , will be a not necessarily orthogonal basis for as long as .
The rows of another matrix are given by the components of three other vectors . By considering the matrix , where denotes the transpose, show that there is a unique choice of such that is also a basis and
Show that the new basis is given by
Show that if either or is an orthonormal basis then is a rotation matrix.