Parallelogram Law: This is a graphical method used for a) addition of two vectors, b) subtraction of two vectors, and c) resolution of a vector into two components in arbitrary directions.
Vector Addition: Consider vectors 
and 
as shown
below. The addition of these two vectors gives the
resultantvector. The following steps are used to find the
resultant vector.
Step 1: As the first step, we draw
a line, at the head of vector , parallel
to vector . We then repeat this for the
other vector.
Step 2: Next, we draw a line from the point of concurrency of the two vectors to the point of intersection of the two parallel lines.
Step 3: Finally, we complete the parallelogram sketch with the diagonal representing the resultant vector.
Notice that in constructing a parallelogram, the two vectors being added have to be shown in a tail-to-tail arrangement.
Vector Subtraction: If we are interested
in subtracting vector from vector , we can represent this operation as the
addition of vectors and (-). Notice that - has
the same magnitude as , but is in
opposite direction.
Step 1: As the first step, we flip
the direction of vector to create vector
-. Then slide it along its axis such that
vectors and -
are tail-to-tail.
Step 2: We then repeat step 1 used in vector addition. We draw a line at the head of each vector parallel to the other vector.
Step 3: The parallelogram law is shown below with the diagonal representing the resultant vector.
Resolution of a Vector into Two Components: We can also use the parallelogram law to determine the components of a vector along any two arbitrary axes. Click the mouse over each step to see the flash animation of this procedure. Also demonstrated is the head-to-tail construction of vector triangles.
Animation of Steps in Resolving a Vector into Two
Components
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