When vectors are added, we need to add both a magnitude **and** a direction. For example, take 2 steps in the forward direction, stop and then take another 3 steps in the forward direction. The first 2 steps is a displacement vector and the second 3 steps is also a displacement vector. If we did not stop after the first 2 steps, we would have taken 5 steps in the forward direction in total. Therefore, if we add the displacement vectors for 2 steps and 3 steps, we should get a total of 5 steps in the forward direction. Graphically, this can be seen by first following the first vector two steps forward and then following the second one three steps forward (ie. in the same direction):

We add the second vector at the end of the first vector, since this is where we now are after the first vector has acted. The vector from the tail of the
first vector (the starting point) to the head of the last (the end
point) is then the sum of the vectors. This is the *head-to-tail* method of vector addition.

As you can convince yourself, the order in which you add vectors does
not matter. In the example above, if you decided to first go 3 steps
forward and then another 2 steps forward, the end result would still be 5
steps forward.

The final answer when adding vectors is called the **resultant**. The resultant displacement in this case will be 5 steps forward.

- Definition 3: Resultant of Vectors
The resultant of a number of vectors is the single vector whose effect is the same as the individual vectors acting together.

In other words, the individual vectors can be replaced by the
resultant – the overall effect is the same. If vectors a→a→ and b→b→ have a resultant R→R→, this can be represented mathematically as,

R
→
=
a
→
+
b
→
.
R
→
=
a
→
+
b
→
.

(1)Let us consider some more examples of vector addition using displacements. The arrows tell you how far to move and in what
direction. Arrows to the right correspond to steps forward, while
arrows to the left correspond to steps backward. Look at all of the
examples below and check them.

This example says 1 step forward and then another step forward is the same as an arrow twice as long – two steps forward.

This examples says 1 step backward and then another step backward is the same as an arrow twice as long – two steps backward.

It is sometimes possible that you end up back where you started. In this case the net result of what you have done is that you have gone nowhere
(your start and end points are at the same place). In this case, your resultant displacement is a vector with length zero units. We use the symbol 0→0→ to denote such a vector:

Check the following examples in the same way. Arrows up the page can be
seen as steps left and arrows down the page as steps right.

Try a couple to convince yourself!

It is important to realise that the directions are not special– `forward
and backwards' or `left and right' are treated in the same way. The same is
true of any set of parallel directions:

In the above examples the separate displacements were parallel to one
another. However the same head-to-tail technique of vector addition
can be applied to vectors in any direction.

Now you have discovered one use for vectors; describing resultant
displacement – how far and in what direction you
have travelled after a series of movements.

Although vector addition here has been demonstrated with
displacements, all vectors behave in exactly the same way. Thus, if
given a number of forces acting on a body you can use the same method
to determine the resultant force acting on the body. We will return to
vector addition in more detail later.