Vectors  (Page 4/10)

What have we done? You originally took 5 steps forward but then you took 3 steps back. That backward displacement would be represented by an arrowpointing to the left (backwards) with length 3. The net result of adding these two vectors is 2 steps forward:

Thus, subtracting a vector from another is the same as adding a vector in the opposite direction (i.e. subtracting 3 steps forwards is the same as adding 3 steps backwards).

In the problem, motion in the forward direction has been represented by an arrow to the right. Arrows to the right are positive and arrows to the left are negative. More generally, vectors in opposite directions differ in sign (i.e. if we define up as positive, thenvectors acting down are negative). Thus, changing the sign of a vector simply reverses its direction:

In mathematical form, subtracting $\overrightarrow{a}$ from $\overrightarrow{b}$ gives a new vector $\overrightarrow{c}$ :

This clearly shows that subtracting vector $\overrightarrow{a}$ from $\overrightarrow{b}$ is the same as adding $(-\overrightarrow{a})$ to $\overrightarrow{b}$ . Look at the following examples of vector subtraction.

Scalar multiplication

What happens when you multiply a vector by a scalar (an ordinary number)?

Going back to normal multiplication we know that $2\times 2$ is just 2 groups of 2 added together togive 4. We can adopt a similar approach to understand how vector multiplication works.

Techniques of vector addition

Now that you have learned about the mathematical properties of vectors, we return to vector addition in more detail. There are a number oftechniques of vector addition. These techniques fall into two main categories - graphical and algebraic techniques.

Graphical techniques

Graphical techniques involve drawing accurate scale diagrams to denote individual vectors and their resultants. We next discuss the two primarygraphical techniques, the head-to-tail technique and the parallelogram method.

The head-to-tail method

In describing the mathematical properties of vectors we used displacements and the head-to-tail graphical method of vector additionas an illustration. The head-to-tail method of graphically adding vectors is a standard method that must be understood.

Method: Head-to-Tail Method of Vector Addition

1. Draw a rough sketch of the situation.
2. Choose a scale and include a reference direction.
3. Choose any of the vectors and draw it as an arrow in the correct direction and of the correct length – remember to put anarrowhead on the end to denote its direction.
4. Take the next vector and draw it as an arrow starting from the arrowhead of the first vector in the correct direction and of thecorrect length.
5. Continue until you have drawn each vector – each time starting from the head of the previous vector. In this way, the vectors to beadded are drawn one after the other head-to-tail.
6. The resultant is then the vector drawn from the tail of the first vector to the head of the last. Its magnitude can bedetermined from the length of its arrow using the scale. Its direction too can be determined from the scale diagram.