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Adding Vectors
To illustrate how to add together two vectors, start with the "simplest" vector, displacement.  
   
It should be clear that an obvious way to add two displacements together is by drawing a scale diagram, as shown below.  
   
   
   
A body moved first from point A to point B.  
   
It then moved from point B to point C.  
   
   
   
   
   
These two displacements result in the body starting at A and finishing at C and therefore can be replaced by displacement s (unless you particularly want to have a look around B on your way to C!)  
   
s is called the resultant displacement.  
   
It is the addition or sum of the two displacements.  
   
Using bold letters to represent vectors the process is represented as  
 
   
   
   
Displacement is a vector quantity.  
This means that to fully describe a displacement, you must state the magnitude, the direction and the sense (up or down, right or left etc) of the displacement.  
   
It would seem reasonable to assume that other vector quantities (eg velocity, acceleration, force etc) might be added using the same method.  
This assumption can easily be verified for forces: see experiment 9M  
   
   
   
Consider two forces represented by the arrows shown here.  
   
We assume that these arrows have been drawn to scale.  
   
   
That is, the length of each arrow indicates the magnitude of each of the forces.  
     
For example, a 1cm length arrow might represent a 10N force.  
 
Drawing the arrows "head to tail", as in the example of adding displacements above, gives us this diagram (sometimes referred to as a triangle of forces... for pretty obvious reasons!)  
   
Measuring the length of the red arrow (and using the scaling factor) gives us the resultant of the two forces.  
   
Again this would be represented symbolically as:  
 
(again, bold characters for vectors).  
   
However, in practice, when we have two forces to add together, they are often acting at the same point  
For this reason, we usually draw the diagram in a slightly different way, in order to make it look more like the situation to which it applies.   
   
First, imagine moving the arrow representing F2 as shown here.  
   
   
   
   
Now, we complete a parallelogram around the forces (this diagram is often called a... guess what... yes, a parallelogram of force!)   
   
The magnitude of the resultant force is now found by measuring the length of the diagonal of the parallelogram.   
   
   
   
This, of course, gives us the same result as the triangle method but has the advantage of looking a bit more like the practical situation to which it refers.   
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