52 Vectors and geometry in three dimensions
of the projection of V on u. In case -Ir/2 < 0 < in, the scalar product is the
negative of this number. The definition of implies that 0 if
and only if cos B = 0. Thus 0 if and only if u and v are orthogonal
(that is, perpendicular to each other). Those who are or want to be con-
versant with principles of physics can note that if a particle P moves from
the tail to the head of the vector u with the constant force v acting upon P
during the motion, then is the work done by the force during the motion.
Referring to Figure 2.181, we can see that if u and v are collinear
vectors (vectors which lie on the same line), then 0 is 0 or in, so sin 0 = 0.
In this case the vector n of the formula (2.17) is not determined, but
u X v is defined to be 0 anyway. Henceforth, we suppose that 0 <
B < ir. In this case, the vector n is the unit normal to the plane of u and
v which is determined by the right-hand rule. A right hand is so placed
that the thumb is perpendicular to the plane of u and v and the fingers are
parallel to this plane and point in the direction that a line rotates in passing
over the geometric angle 0 from u to v (not v to u). The unit normal n
is then the vector which has the direction of the thumb and which is one
unit long. From Figure 2.181 we see that lvJ sin 0 is the altitude of the
triangle of which the vectors u and v form two sides. It follows from
(2.17) that u X v = 2An, where A is the area of this triangle. It must
always be remembered that the vector product u x v is a vector which,
when it is not 0, has the direction of the thumb when the right-hand rule
is applied. Moreover, it is necessary to observe and remember that,
except when u x v = 0, the vector v x u is not the same as the vector
u x v. After having found u X v by the right-hand rule, we must flip
the hand over so that the thumb points in the opposite direction to find
v x u, and it follows that(2.182) v X u = -u X V.
Anyone can attain complete understanding of these matters by making a
few experiments in which two pencils (representing vectors) are held in
the left hand while the right hand is used to determine the direction of
their vector product. While vector products appear infrequently in this
book, they have many important applications.
Finally, we call attention to some simple formulas that are easy to use
but are not so easy to prove. The basic formula, which is proved in
Problem 17 below, is(2.183) (u1 + v2) = ur(v1 + v2) + v2)
u1*v1 + u1'v2 +Analogous formulas hold when the parentheses in the left member contain
sums of more than two vectors. Moreover, correct formulas are obtained
by replacing the dots by crosses. Proofs of this fact are given in text-
books on vector analysis.