to is called the vector projection of on , and the component perpendicular to is called the vector rejection of from . The vector can be viewed as a copy of rotated anticlockwise by 90° about , so their magnitudes are equal but directions are perpendicular. Likewise the vector a copy of rotated anticlockwise through about , so that and are equal in magnitude but in opposite directions (i.e. they are negatives of each other, hence the minus sign). Expanding the vector triple product establishes the connection between the parallel and perpendicular components, for reference the formula is given any three vectors , , . The component parallel to the axis will not change magnitude nor direction under the rotation, only the perpendicular component will change direction but retain its magnitude, according to and since and are parallel, their cross product is zero , so that and it follows This rotation is correct since the vectors and have the same length, and is rotated anticlockwise through about . 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 Matrix notation Representing and as column matrices, the cross product can be expressed as a matrix product By , denote the "cross-product matrix" for the unit vector , That is to say, for any vector . (In fact, is the unique matrix with this property. It has eigenvalues 0 and ). It follows that iterating the cross product is equivalent to multiplying by the cross-product matrix on the left; specifically: The previous rotation formula in matrix language is therefore So we have: 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 generating that Lie group (note that is skew-symmetric, which characterizes ). In terms of the matrix exponential, To see that the last identity holds, one notes