Vector addition
When two or more vectors are added the resulting sum of the vectors is called the RESULTANT
vector or simply the RESULTANT. This could be a resultant velocity, force, acceleration etc.
depending on the nature of the original vectors.
We will consider the addition of two or more vectors first when they all act in the same line and
then when they act at angles to each other.
(i) vectors acting in the same line
Two or more vectors acting in the same direction may be added as if they were scalars. For
example the sum, or resultant of the three forces shown in Fig. 1(a) is 50 N acting right to left while
in (b) it is 250 N left to right.
100N
100N
150N
150N
250 N
50 N
Figure 1(a)
Simple vector addition
Figure 1(b)
(ii) vectors acting in different directions
However if the two vectors are not acting along the same line the triangle of vectors shown below
can be used to add them together.
Using the same magnitude for the two vectors as those already
considered we draw a scale diagram in both magnitude and
direction as shown in Figure 2. The resultant (R) (= 214N in
this case) is the vector that closes the triangle.
214N
150N
100N
Notice that the original two vectors (shown blue in the diagram) follow each other round the
triangle (nose to tail) to give the resultant, the red vector (R), and that this resultant acts in the
opposite direction round the triangle.
A further example is shown in Figure 3.
The two vectors (A and B) that you
wish to add are represented in
magnitude and direction by a scale
diagram as before (see Figure 3).
In this example A = 73 N, B = 40 N
and the resultant (R) is then found
by measuring the closing vector
and is found to be 62 N.
Addition of two vectors
72 N
B
40 N
A
Figure 3
R
62 N
1
If more than two vectors act at a
point as in Figure 5(a) then the
polygon of vectors can be used.
The resultant is still the vector
(shown red) that closes the polygon.
(See Figure 5(b)). The original
vectors follow nose to tail around
the polygon while the resultant
faces the opposite way.
resultant
Figure 5(b)
Figure 5(a)
Walking in the rain – vector addition
A man walking through a rainstorm is a good
example of the addition of vectors (see Figure 6).
If he walks at 1.5 m/s and the rain is falling
vertically at 2 m/s then the rain will be hitting him
at 2.5 m/s at an angle of 37o to the vertical.
Figure 6
Aircraft and a cross wind
The next example of vector addition shows an
aircraft flying on an initial bearing of 0o at 350 ms-1
with a wind blowing west-east at 50 ms-1.
(See Figure 7)
The resulting velocity of the plane compared with
the ground is 354 ms-1 on a bearing of N 8.1o E.
Wind blowing west-east
(bearing 270o)at 50 ms-1
To find the resultant of the
motion of the aircraft and
the wind the two vectors
are drawn nose to tail in a
scale
diagram.
The
resultant is the line that
completes the diagram.
Final path of
aircraft is N
8.1o E moving
with a speed
of 354 ms-1
Figure 7
Aircraft flying south-north (bearing 0o) at 350 ms-1
2