The Head-to-Tail Method

This method has advantages in being easy to apply, especially when multiple vectors are to be summed. To sum vectors using this method, simply move them such that the head of one vector is attached to the tail of the other. Once all the vectors have been "chained together," the resultant vector is easy to discern -- it is simply a vector that points from the start of the chain to the end. The following applet demonstrates this method very well for any two vectors.

Can we really move vectors around like that? What are the rules for moving vectors?

Applet by B. Surendranath Reddy

Using the mouse, create two vectors. Notice that when moving a vector its length and direction do not change. Notice what happens when the tail of one vector is placed on the head of another -- the resultant vector appears. Note that this resultant vector points from the tail of the first vector to the head of the second, just as we described earlier.

Notice that the parallelogram method shares the same basic disadvantage as the head-to-tail method -- finding the exact length and direction of the resultant vector is going to again require some sophisticated trigonometry. Therefore, this method is also used mainly to provide a general, qualitative description of the sum of one or more vectors.

Special Examples of Vector Summation

This is a good point to demonstrate two important special cases of vector algebra. In the above applet, orient one of the vectors so that it points in the same direction as the other.

When two vectors point in the same direction, (1) the resultant vector points in the same direction, and (2) the length of the resultant vector is simply the sum of the lengths of the individual vectors.

Now clear the screen and draw the second vector so that it points in the opposite direction of the first.

When two vectors point in opposite directions, (1) the length of the resultant vector is found by subtracting the length of the smaller vector from the length of the larger vector, and (2) the resultant vector points in the direction of the larger vector.

Test these two principles with the first applet shown on this page.

Multiple Vector Summation

One important advantage of this method over the parallegram method (discussed next) is the ease in which multiple vectors can be added. Using the following applet, add three or more vectors together. Easy, heh?

Applet by Walter Fendt

Notice one distinct disadvantage of this method -- finding the exact length and direction of the resultant vector is usually going to require some sophisticated geometry, especially when summing multiple vectors. Therefore, this method is used mainly to provide a rough description of the sum of one or more vectors.