Mathematics of Physics and Engineering

18 Vector Operations

Definition 1.4 Let u and v be two vectors in R^3. Let 8 be the angle
between tt and v (0 < 9 < IT, see Figure 1.2.1). The cross product, u x v,
is the vector having magnitude \u x v\ = ||u||.||u|| sin# and lying on the
line perpendicular to u and v and pointing in the direction in which a
right-handed screw would move when u is rotated toward v through angle
6.

Sometimes, the symbol Q is used to represent a vector perpendicular
to the plane and coming out of the plane toward the observer, while the
symbol ® represents a similar vector, but going away from the observer;
see Figure 1.2.6.
The triple (u,v,u x v) forms a right-handed triad (Figure 1.2.5). More
generally, we say that an ordered triplet of vectors (u, v, w) with a common
origin in R^3 is a right-handed triad (or right-handed triple) if the vectors
are not in the same plane and the shortest turn from u to v, as seen from
the tip of w, is counterclockwise.

U X V

JC.

u

U X V

Fig. 1.2.5 The Cross Product I

o- u

U X V U X V

Fig. 1.2.6 The Cross Product II

An important application of cross-product in mechanics is the moment
of a force about a point O. Suppose an object located at a point P is
subjected to a force vector F, applied at P. Let r be the position vector of
P. The force F tends to rotate the object around O and exerts a torque,
or moment, T around O. (The Latin verb torquere means "to twist.") The
magnitude of the torque T is ||T|| = ||r||.||F||sin0, where 6 is the angle
between r and F; recall that a.b denotes the usual product of two numbers