Understanding Engineering Mathematics

11.12 Vector functions

Avector functionis a vector which is a function of some variable, sayt. We writef(t)
for a general vector function. For example the position of a projectile may be represented
by a vector,rreferred to some origin, and this position will vary with time,t. We can
represent this by takingrto be a function oft,r(t).Sor(t)is a vector function of time
t. This is illustrated in Figure 11.16.

P x(t),y(t),z(t)

x

y

z

r (t) = x(t)i + y(t)j + z(t)k

Figure 11.16Position vector of a vector function.

The set of values of the variabletis called thedomainof the vector function, while
the set of possible vectorsf(t)is called thecodomainorrangeof the vector function.
A vector for which both the magnitude and direction are constant, i.e. each component
is constant, is called aconstant vector. Examples of constant vectors arei,j,k.
It is useful to know the vector functions representing lines and planes. The position
vectorrof a general pointPlying on a straight line passing through a point with position
vectorais

r=a+td

wheredis any constant vector parallel to the line – see Figure 11.17.

O

r = a + td

d td

a

Figure 11.17Equation of a line.

For the case of a plane, consider Figure 11.18.
LetP be any point in a plane, and letnbe any vector normal to the plane. Letabe
the position vector of a fixed point in the plane. Ifris the position vector ofPthenr−a
will be a vector in the plane. As such it will be perpendicular ton, from which we have

(r−a)·n= 0

So
r·n=a·n=ρsay