3 MOTION IN 3 DIMENSIONS 3.8 Vector velocity and vector acceleration
centroid X from point A can be written in one of two different ways:
x (^) = a + λ d, (3.9)
x = b + a − μ c. (3.10)
Equation (3.9) is interpreted as follows: in order to get from point A to point
X, first move to point B (along vector a), then move along diagonal BD (along
vector d) for an unknown fraction λ of its length. Equation (3.10) is interpreted
as follows: in order to get from point A to point X, first move to point D (along
vector b), then move to point C (along vector a), finally move along diagonal CA
(along vector −c) for an unknown fraction μ of its length. Since X represents the
same point in Eqs. (3.9) and (3.10), we can equate these two expressions to give
a + λ (b − a) = b + a − μ (a + b). (3.11)
Now vectors a and b point in different directions, so the only way in which the
above expression can be satisfied, in general, is if the coefficients of a and b
match on either side of the equality sign. Thus, equating coefficients of a and b,
we obtain
1 − λ = 1 − μ, (3.12)
λ = 1 − μ. (3.13)
It follows that λ = μ = 1/2. In other words, the centroid X is located at the
halfway points of diagonals BD and AC: i.e., the diagonals mutually bisect one
another.
1.21 Vector velocity and vector acceleration
Consider a body moving in 3 dimensions. Suppose that we know the Cartesian
coordinates, x, y, and z, of this body as time, t, progresses. Let us consider how
we can use this information to determine the body’s instantaneous velocity and
acceleration as functions of time.
The vector displacement of the body is given by
r(t) = [x(t), y(t), z(t)]. (3.14)