Now that you have learned about the mathematical properties of vectors, we return to vector addition in more detail. There are a number of techniques of vector addition. These techniques fall into two main categories  graphical and algebraic techniques.
Graphical techniques
Graphical techniques involve drawing accurate scale diagrams to denote individual vectors and their resultants. We will look at just one graphical method: the headtotail method.
Method: HeadtoTail Method of Vector Addition

Draw a rough sketch of the situation.

Choose a scale and include a reference direction.

Choose any of the vectors and draw it as an arrow in the correct direction and of the correct length – remember to put an arrowhead on the end to denote its direction.

Take the next vector and draw it as an arrow starting from the arrowhead of the first vector in the correct direction and of the correct length.

Continue until you have drawn each vector – each time starting from the head of the previous vector. In this way, the vectors to be added are drawn one after the other headtotail.

The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can be determined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram.
Let's 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 step forward and then another step forward is the same as an arrow twice as long – two steps forward.
This example says 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
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 headtotail technique of vector addition can be applied to vectors in any direction.
Worked example 3: Headtotail addition 1
Draw a rough sketch of the situation
Worked example 4: Headtotail addition 2
Algebraic techniques
Vectors in a straight line
Whenever you are faced with adding vectors acting in a straight line (i.e. some directed left and some right, or some acting up and others down) you can use a very simple algebraic technique:
Method: Addition/Subtraction of Vectors in a Straight Line

Choose a positive direction. As an example, for situations involving displacements in the directions west and east, you might choose west as your positive direction. In that case, displacements east are negative.

A tennis ball is rolled towards a wall which is away from the ball. If after striking the wall the ball rolls a further along the ground away from the wall, calculate algebraically the ball's resultant displacement.
Draw a rough sketch of the situation
Choose a positive direction
Let's choose the positive direction to be towards the wall. This means that the negative direction is away from the wall.
Now define our vectors algebraically
With right positive:
Add the vectors
Next we simply add the two displacements to give the resultant:
Quote the resultant
Finally, in this case towards the wall is the positive direction, so: towards the wall.
Worked example 6: Subtracting vectors algebraically 1
Draw a sketch
A quick sketch will help us understand the problem.
Decide which method to use to calculate the resultant
Remember that velocity is a vector. The change in the velocity of the ball is equal to the difference between the ball's initial and final velocities:
Since the ball moves along a straight line (i.e. left and right), we can use the algebraic technique of vector subtraction just discussed.
Choose a positive direction
Choose the positive direction to be towards the wall. This means that the negative direction is away from the wall.
Now define our vectors algebraically
Subtract the vectors
Thus, the change in velocity of the ball is:
Quote the resultant
Remember that in this case towards the wall means a positive velocity,
so away from the wall means a negative velocity: away from the wall.
Worked example 7: Adding vectors algebraically 2
Draw a sketch
A quick sketch will help us understand the problem.
Decide which method to use to calculate the resultant
Remember that force is a vector. Since the crate moves along a straight line (i.e. left and right), we can use the algebraic technique of vector addition just discussed.
Choose a positive direction
Choose the positive direction to be towards the crate (i.e. in the same direction that the man is pushing). This means that the negative direction is away from the crate (i.e. against the direction that the man is pushing).
Now define our vectors algebraically
Subtract the vectors
Thus, the resultant force is:
Quote the resultant
Remember that in this case towards the crate means a positive force: towards the crate.
Remember that the technique of addition and subtraction just discussed can only be applied to vectors acting along a straight line. When vectors are not in a straight line, i.e. at an angle to each other then simple geometric and trigonometric techniques can be used to find resultant vectors.