Adding vectors & scalars

Scalars

adding two or more scalars quantities is a simple thing and can be done by simple law of addition .

like in this  example. A man walks 3km east and 4km north thus his total distance is
3km + 4km= 7km
as distance is a scalar quantity so simple law of addition can be used .

Vectors

1. 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.

Similarly orientated vectors can be subtracted the same manner.

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.

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.

2. 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 (see above on the right).

3. 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:

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

4. 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.

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.

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

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

5. 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.

The magnitude of A, |A|, can be calculated from the components, using the Theorem of Pythagoras:

and the direction can be calculated using

Note that if a vector is directed along one of the axes, then the component along the other axis is zero.