ADDITION OF VECTORS

Contents for this page	Related topics
Addition of vectors Polygon method Parallelogram method Method of components Rectangular components Additional questions	Introduction to vectors Physical examples of vectors	Data Glossary
Learning Outcomes
After studying this section, you will be able to add vectors by the polygon method, the parallelogram method, and the method of components.

Addition of vectors:

Two or more vectors may be added together to produce their ADDITION. If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction.

First illustration of what is meant by a resultant

Similarly orientated vectors can be subtracted the same manner.

Second illustration of what is meant by a resultant

It follows that vectors can also be multiplied by a scalar, so for example if the vector A were multiplied by the number m, the magnitude of the vector, |A|, would increase to m|A|, but its direction would not change.

Illustration of a vector multiplied by a scalar

In general, since vectors may have any direction, we must use one of three methods for adding vectors. These are, the POLYGON METHOD, PARALLELOGRAM METHOD and the METHOD OF COMPONENTS.

Polygon method:

Two vectors A and B are added by drawing the arrows which represent the vectors in such a way that the initial point of B is on the terminal point of A. The resultant C = A + B, is the vector from the initial point of A to the terminal point of B.

Many vectors can be added together in this way by drawing the successive vectors in a head-to-tail fashion, as shown here on the left.

If the polygon is closed, the resultant is a vector of zero magnitude and has no direction. This is called the NULL VECTOR, or 0.

Parallelogram method:

In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin and the parallelogram constructed as follows:

Illustration of the parallelogram method of vector addition

The resultant R is the diagonal of the parallelogram drawn from the common origin.

Method of components:

The components of a vector are those vectors which, when added together, give the original vector.

The sum of the components of two vectors is equal to the sum of these two vectors.

Normal components If components are appropriately chosen, this theorem can be used as a convenient method for adding vectors.

The direction of vectors is always defined relative to a system of axes. For example, in discussing displacement on the surface of the earth, it is convenient to use axes directed from South to North and from West to East:

In such a situation, an arbitrary displacement A can be thought of as being made up of two components A1 and A2 directed along these axes, such that A = A1 + A2.

A vector as the sum of two component vectors

The components could be determined by constructing a scale diagram, but they are easily calculated as follows (Note: It is convenient to specify Θ clockwise from North when referring to displacements on the earth:

Normal components of a vector

A1, the component in an easterly direction, will have a magnitude |A1| = |A| cosΘ.
A2, the component in a northerly direction, will have a magnitude |A2| = |A|sinΘ

Rectangular components:

In all vector problems a natural system of axes presents itself. In many cases the axes are at right angles to one another. Components parallel to the axes of a rectangular system of axes are called rectangular components.

In general it is convenient to call the horizontal axis X and the vertical axis Y. The direction of a vector is given as an angle counter-clockwise from the X-axis.