Definition of a Vector  Review of Plane Trigonometry
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 tailtotail 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 tailtotail.
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 headtotail construction of vector triangles.
Animation of Steps in Resolving a Vector into Two
Components
