Understanding Engineering Mathematics

q

d

F

Figure 11.13Force at an angle to the direction of motion.

wire). This component isFcosθ. So, if we move the bead a distancedalong the wire
then the work done is

W=(Fcosθ)d

=Fdcosθ

This is the generalisation of equation (1) when the force and displacement make an angle
θ with each other. So, letFbe the vector representing the force anddrepresent the
displacement. Then the work done is:

W=|F||d|cosθ

i.e. the magnitude of the forceFmultiplied by the magnitude of the displacementd, times
the cosine of the angle between the two vectors.
The result is, naturally, a scalar. But it has a magnitude component,|F||d|,anda
direction componentθ, which suggest that it would be very useful to define this particular
combination of vectors as a ‘product’ of vectors which yields a scalar. In general then the
scalar productof two vectorsa,bis denoted bya·b(sometimes called thedot product)
and defined by

a·b=abcosθ

wherea=|a|,b=|b|andθis the angle betweenaandb– see Figure 11.14.

q

b

a

Figure 11.14Thescalarproductisa·b=abcosθ.

With this definition the work done by the forceFmoving a particle through a displace-
mentdis given by

W=F·d

This explains why we define the scalar product as above – it is very useful to do so!
Now, for the purposes of calculations it is invariably more convenient to have an expres-
sion for the scalar product in terms of components. The definition we have given above,
as we will see shortly, is equivalent to the following expression in terms of components:

a·b=a 1 b 1 +a 2 b 2 +a 3 b 3 =b·a