An appropriate scaling of and using the trigonometric functions sine and cosine gives the rotated perpendicular component. The form of the rotated component is similar to the radial vector in 2D planar polar coordinates in the Cartesian basis   where , are unit vectors in their indicated directions. Now the full rotated vector is  By substituting the definitions of and in the equation results in  Representing and as column matrices, the cross product can be expressed as a matrix product  Letting denote the "cross-product matrix" for the unit vector , the matrix equation is, symbolically, for any vector . (In fact, is the unique matrix with this property. It has eigenvalues 0 and ). Iterating the cross product on the right is equivalent to multiplying by the cross product matrix on the left, in particular  Moreover, since is a unit vector, has unit 2-norm. The previous rotation formula in matrix language is therefore  Note the coefficient of the leading term is "now" 1, in this notation: see the Lie-Group discussion below. Factorizing the allows the compact expression where   is the rotation matrix through an angle counterclockwise about the axis , and the identity matrix. This matrix is an element of the rotation group of , and is an element of the Lie algebra formula_18 generating that Lie group (note that is skew-symmetric, which characterizes formula_18). In terms of the matrix exponential,  To see that the last identity holds, one notes that characteristic of a one-parameter subgroup, i.e. exponential, and that the formulas match for infinitesimal . For an alternative derivation based on this exponential relationship, see exponential map from formula_18 to. For the inverse mapping, see log map from to formula_18. Note that the Hodge dual of the rotation formula_24 is just formula_25 which allows the extraction of both the axis of rotation and the sine of the angle of the rotation from the rotation itself, with the usual ambiguity:  where formula_28. The above simple expression results from the fact that the Hodge dual of formula_29 and formula_30 are zero, and formula_31.