Mathematics of Physics and Engineering

Cross Product 19

a, b. The quantity ||F|| sin# is the magnitude of the component of F per-

pendicular to r. (The component of F along r has no rotational effect.)

The magnitude ||r|| is called the moment arm. Our experience with levers

convinces us that the torque magnitude is proportional to the moment arm

and the magnitude of force applied perpendicular to the arm. Hence, we

define the torque of F around O to be the vector T = r x F, where r is

the position at which F is applied. The direction of T is perpendicular to

r and F and (r, F, T) is a right-handed triad.

PROPERTIES OF THE CROSS PRODUCT. From the definition it follows

immediately that the vector w = u xv has the following three properties:

(CI) H| = ||«||.|H|sin(?.
(C2) w • u = w • v = 0.
(C3) —w = v x u.

A fourth property captures the geometry of the right-handed screw in

algebraic terms. Choose any right-handed cartesian coordinate system

given by three orthonormal vectors i, j, k. Suppose the components of

the vectors u, v, w = u x v in the basis (z, j, k) are, respectively,

(ui,U2,u 3 ), (vi,V2,v 3 ),&nd(wi,W2,w 3 ). Then

(

wi u 2 u 3 \

vi v 2 v 3 >0,

Wx W 2 W 3 J

where det is the determinant of the matrix; a brief review of linear algebra,

including the determinants, is in Appendix. To prove (C4), choose k' =

w/\\w\\, j' = v/||v||, and select a unit vector %' orthogonal to both k'

and j' to make (?', j', k') a right-handed triad. In this new coordinate

system, property (C4) becomes

(

u[ u' 2 0 \

0 ||v|| 0 =ui||t>||.|H| >0. (1.2.15)

0 0 HI/

Since (u,v,w) is a right-handed triad, the choice of %' implies that u[ >

0, and (1.2.15) holds. For the system z, j, k with the same origin as

(?', j', k'), consider an orthogonal transformation that moves the basis

vectors i, Vcj, k to the vectors ?', Vcj', k', respectively. If B is the matrix

representing this transformation in the basis (i, Vcj, k), then deti? = 1,