**Summary**

- A unit vector is the vector whose magnitude is 1 unit. It is used to specify the direction of the given vector.
- Unit vector is defined as:
- Magnitude of a vector is:
- A vector can be further broken down or resolved into its two components: vertical and horizontal. Vertical component is defined as and horizontal component is defined as .

Vector quantities are extremely useful in physics. The important characteristic of a vector quantity is that it has both a magnitude and a direction. Both of these properties must be given in order to specify a vector completely.

#### Unit vector

A unit vector is the vector whose magnitude is 1 unit. It is used to specify the direction of the given vector. Also:

i.e

If a vector is divided by its magnitude (modulus) then we get a unit vector in the direction of that vector.

Unit vectors can be described as *i + j+ k*, where *i* is the direction of the *x axis* and *j* is the direction of the *y axis* and *k* is the direction of the* z axis*.

If *r = ai +bj+ck*, then the unit vector in the direction of *r* is given by:

The denominator of the above formula is basically the magnitude of the vector, therefore to work out the magnitude of this vector, we use the formula:

#### Addition of Vectors

Given two vectors a and b, we form their sum a + b, as follows.

We translate the vector b until its tail coincides with the head of a.

Then, the directed line segment from the tail of a to the head of b is the vector a+b just as shown in Fig 1.

#### Example #1

Q. Find the sum of vectors *p = 2i + 5j +3k* and *q = i – 2j + 2k*.

*Solution:*

We carry out the addition of two vectors in the following way:

*p + q = (2 + 1)i + (5 – 2)j + (3 + 2)k*

*p + q = 3i + 3j +5k*

#### Example #2

Q. Work out *a – b* when *a = 10i + 12j – 7k* and *b = 5i – 2j + 8k*

*Solution:*

*a – b = (10 – 5)i + (12 – (-2))j + (-7 – 8)k*

*a – b = 5i+ 10j- 15k*

#### Resolving a Vector

A vector can be further broken down or resolved into its two components; vertical and horizontal.

Let’s take an example of a vector in figure 2 (Fig 2)

This vector can be resolved into two components:

1. Vertical component which is along the y axis.

2. Horizontal component which is along the x axis

We can calculate the vertical and horizontal components if we know the magnitude and direction of the vector.

In other words: we can work out the components of the vector if we know the length of the line and the angle between it and the horizontal or vertical axis.

Let’s suppose vector *r* in Fig 2, its vertical component is defined as , and its horizontal component is defined as .

Therefore:

#### Example #3

Q. Find vector* r* given that magnitude of and angle between the vector and its horizontal component is 30°.

*Solution:*

We resolve the vector into two components as shown in Fig 3 below.

Therefore:

- vertical component:

- horizontal component:

Hence:

*r = 5i + 8.66 j*