# Vectors

###### Sections

##### Resultant displacement

Imagine that after going from point A to point B, our object proceeds onto point C, taking the curved path shown in Figure 4.

Figure 4

The second displacement is also represented by an arrow, as shown. Let us call the first displacement **a** and the second one, **b**. Vectors are written in bold.

Figure 5

The object went from A to C, via B. Let us call the resulting displacement **c**. It points from A to C. **c** is the result of **a** followed by **b**. Vector **c** is called the resultant of vectors **a** and **b**.

We write it symbolically as,

(1)

The plus sign stands for the operation *place second vector so that its tail coincides with the head of the first, then join the tail of the first to the head of the second to get the resultant*.

Ok. You might want to read that sentence again, slowly. Look at the drawing above to see it. It is not a plus in the usual sense.

This is called Vector Addition. This is how vector addition is defined. All vectors are added in this fashion.

We could have used any symbol to represent this operation on two vectors to get a third, but the wise ones have chosen to use a '+' since (as we will see) the operation has all the properties of usual addition. Also note that we use boldface letters to denote vectors. While writing in longhand, place an arrow on top of the letter to denote a vector:

**Q**. What is the sum of two vectors, a 2 unit long vector pointing north and another 2 unit long one pointing west?

**A**. A 2√2 unit long vector pointing northwest.

Figure 6

The magnitude of the first vector is 2. We write |**a**| = 2. The magnitude of the second vector is also 2: |**b**| = 2. The magnitude of the resultant **c**, is 2√2: |**c**| = 2√2. Note that the magnitude of the resultant is not the sum of the magnitudes.

**Q**. Can the sum of two non zero vectors be a vector of zero length?

**Q**. Show that vector addition is commutative. That is, **a** + **b** = **b** + **a**. In the case of displacements, it means that performing displacement **a** followed by displacement **b** gives us the same resultant as performing the displacements in the reverse order.