Understanding Engineering Mathematics

11.3 Addition and subtraction of vectors

Geometrically, vectors areaddedby the use of thetriangle lawin whichaandbform
two sides of a triangle in order, as shown in Figure 11.2. Then the resultant is the third
side, denoted here with a double arrow. This representation of addition is in keeping with
vectors describing displacements for example – following vectoraand then continuing
along vectorbtakes you to the same place that the resultant of vectorsaandbtakes you
directly. In Figure 11.2 we represent this explicitly by showing displacements between
three pointsA,B,C.

b

a

a + b

B

A

AB + BC = AC

C

→ → →

Figure 11.2Triangle addition of vectors.

The resultant ofaandbis denoted by the vector suma+bor by

−→
AC. This can be
regarded as ‘afollowed bybgivesa+b’.
An alternative means of geometrical addition, linked to the calculation of the resultant
of two forces, is theparallelogram law. Thus to add vectorsaandbwe construct
a parallelogram withaandbforming two sides and then theirsumorresultant,in
magnitude and direction, is represented by the diagonal of the parallelogram, as shown
in Figure 11.3. This represents the way in which vector quantities such as force combine.
For example, if you imagine two horses on either side of a canal pulling a barge, then
the resultant force on the barge will be represented by the diagonal of the corresponding
parallelogram.

a + b

a

b

Figure 11.3Parallelogram addition of vectors.

Using either viewpoint – triangle or parallelogram – by constructing appropriate
diagrams you can convince yourself that

a+(b+c)=(a+b)+c (vector addition isassociative)

a+b=b+a (vector addition iscommutative)

Three or more vectors can be added by thepolygon lawas illustrated in Figure 11.4 – the
resultant or sum ‘closes’, in the opposite direction, the polygon formed by following the
vectors in the sum in turn.