# Vector Analysis

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 Choose a desired set of components by clicking on the corresponding icon Click on steps 1 through 6 to see the procedure in finding the two components of the vector