= U2V2W2 - U2(V • W)2 - V2(U • W)2 - W2(U • V)2

The following identity provides a means of writing the vector W in terms of U, V, andUxV, if UxV^O:

If A is a real orthogonal matrix,

where the positive sign holds if A is proper, and the negative sign if A is improper.

The tangent of the rotation angle from V to W about U (the angle of the rotation in the positive sense about U that takes VxU into a vector parallel to WxU) is

The quadrant of @ is given by the fact that the numerator is a positive constant multiplied by [email protected], and the denominator is the same positive constant multiplied by cos0. If U, V, and W are unit vectors, @ is the same as the rotation angle on the celestial sphere defined in Appendix A. Equation (C-95) is derived in Section 7.3. (See Eqs. (7-57).)


1. Bellman, R. E., Introduction to Matrix Algebra. New York: McGraw-Hill, Inc., 1960.

2. Carnahan, B., H. A. Luther and J. O. Wilkes, Applied Numerical Methods. New York: John Wiley & Sons, Inc., 1969.

3. Forsythe, George E., and C. Moler, Computer Solution of Linear Algebraic Systems. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1967.

4. Halmos, P. R., ed., Finite-Dimension Vector Spaces, Second Edition. Princeton, NJ: D. Van Nostrand Company, Inc., 1958.

5. Hoffman, Kenneth and Ray Kunze, Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1961.

6. Noble, Ben, Applied Linear Algebra. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1969.

7. Stewart, G. W., Introduction to Matrix Computations. New York: Academic Press, Inc., 1973.

8. Wilberg, Donald M., State Space and Linear Systems, Schaum's Outline Series, New York: McGraw-Hill, Inc., 1971.

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