Understanding Engineering Mathematics

2. (i)

√

21

2

(ii)

1

√

21

(i+ 4 j− 2 k) (iii)− 3 i− 12 j+ 3 k (iv)

1

2

i+ 2 j+k

(v)

5

2

i+ 10 j− 4 k

11.10 The scalar product of two vectors

So far we have simply added or subtracted vectors, or formed linear combinations of them.
An obvious question is what sorts of ‘products’ can we define for two vectors. By ‘product’
we mean a generalisation of the multiplication of two real numbers. It is important to
realise that wearefree todefinesuch products – vectors are new mathematical objects,
completely different from anything we have considered before in this book, and we are at
liberty to define the combinations of them in any way we wishwhich is mathematically
consistent. When we invent new mathematical tools this is always the case – so long as
our definitions do not lead to any silly contradictions (such as ‘1=0’) we can define the
rules exactly as we wish.Butof all the possible rules, we are naturally going to choose the
most useful. So if we do define products of vectors then we are going to do so in such a
way that the result is useful to us in modelling reality. We have (at least) two possibilities
for defining ‘multiplication’ of vectors:

vector∗vector=scalar

vector∗vector=vector

Not surprisingly these give rise to the ‘scalar’ and ‘vector product’ respectively.
The best way to introduce the scalar product is to do a little ‘work’. Specifically, if a
force of magnitudeF acting in a fixed direction moves a particle through a displacement
din that direction, then we say that the ‘work done’ by the force on the particle is:

W=Fd ( 1 )

Now, both force and displacement are vector quantities – they require both magnitude and
direction to specify them. This is hidden in the above result because both the force and the
displacement have been taken as operating in the same fixed direction. So, the right-hand
side of the above equation is essentially a vector times a vector.
On the other hand, work done, or energy expended, is not a vector quantity – energy is
simply a scalar, numerical quantity, like temperature or heat. So the left-hand side of the
equation is a scalar. Thus, the equation essentially says:

“vector ‘×’ vector=scalar”

But, as we said, the vector content is hidden. To bring it out, supposeF no longer acts
on the particle in its own direction, but at an angle to it. For example the particle may be
a bead on a straight smooth wire, and the force applied by pulling a string at an angleθ
to the wire, as shown in Figure 11.13.
Only the component of the force along the direction of the wire can do work on the
bead (the component perpendicular to the wire will not result in any motion along the