Mathematics of Physics and Engineering

24 Curves in Space

Thus,

u • (v x w) = w • (u x v) = (u x v) • w. (1.2.30)

In other words, the scalar triple product does not change under cyclic per-
mutation of the vectors or when • and x symbols are switched.
EXERCISE 1.2.15? Verify that the ordered triplet of non-zero vectors u, v, w
is a right-handed triad if and only if (u, v, w) > 0.
Recall that \v x w\ = ||u|| • ||iu|| sin# is the area of the parallelogram
formed by v and w. Therefore, \u • (v x w)\ is the volume of the paral-
lelepiped formed by u,v, and w. Accordingly, (u,v,w) = 0 if and only
if the three vectors are linearly dependent, that is, one of them can be ex-
pressed as a linear combination of the other two. Similarly, four points
Pi, i = 1,..., 4 are co-planar (lie in the same plane) if and only if

(P 1 P 2 ,PiP 3 ,P 1 P 4 ) = 0, (1.2.31)

where PjP, = OPj — OPi. If (XJ, yi, z^ are the cartesian coordinates of the
point Pi, then (1.2.31) becomes

det

x 2 ~xiy 2 - 2/i z 2 - zi

%3 - x\ 2/3 - 2/i z 3 - zi

x 4 -xiy 4 - j/i z 4 - zx

(1.2.32)

Notice a certain analogy with (1.2.28) and (1.2.29).
EXERCISE 1.2.16.C Let u = (1,2,3), t; = (-2,1,2), w = (-1,2,1). (a)
Compute uxv, vxw, (uxv)x(vxw). (b) Compute the area of the paral-
lelogram formed by u and v. (c) Compute the volume of the parallelepiped
formed by u, v, w using the triple product (u, v, w).

1.3 Curves in Space

1.3.1 Vector-Valued Functions of a Scalar Variable

To study the mathematical kinematics of moving bodies in M^3 , we need to

define the velocity and acceleration vectors. The rigorous definition of these

vectors relies on the concept of the derivative of a vector-valued function

with respect to a scalar. We consider an idealized object, called a point

mass, with all mass concentrated at a single point.