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**Week 1.**

**Performance objectives**

Students should be able to

- State difference between vectors and scalar
- Perform simple operations on vectors;
- Determine the sum, difference of any combination of vectors lying in a plane
- Resolve a vector in a given direction.

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**Scalars and Vectors.**

There are two kinds of physical quantities:

- Those which has only magnitude;
- Those which have both magnitude and direction.

Those which have only magnitude but no direction are called **scalars** while those which have both magnitude and also direction are called **vectors.**

Examples of scalars are length, area, volume, mass, work, energy, speed and density. While examples of vectors are displacement, velocity acceleration, force and momentum.

Vectors may necessarily have any particular position associated with them. Such vectors are usually called **free vectors**. Vectors which have particular positions associated with them may be located along a straight line or through a particular point.

This is a vector:

A vector has **magnitude** (size) and **direction**: The length of the line shows its magnitude and the arrowhead points in the direction.

**Vector Notation:** In printed work, a vector is shown in heavy print e.g. **a **or **b**. In written work, a vector is shown by a letter with a bar underneath it. e.g**. **

**Magnitude of a Vector.** The magnitude of a vector **a, **sometimes called the **Modulus** of the vector is represented by

**The Zero Vector.** The zero vector is a vector with a magnitude of zero. The zero vector has no particular direction. It is represented by **0.** The zero vector is sometimes called **null vector**

**The Unit Vector.** The unit vector in the direction of the vector **a** is the vector represented by **a **and is such that

**The Negative Vector.** The negative vector of a vector **a**, is the vector which has the same magnitude as that if ** ****a **but a direction opposite to that of **a. **The negative vector of **a** is written –a

**Equality of Vectors.** Two vectors are **a** and **b** said to be equal if they have the same magnitude and direction.

**Addition of vector**

Consider the diagram below.

C

P+q q

A p B

From the diagram, the journey from A to C can be accomplished in twi ways;

- From A to B and then from B to C
- From A to C directly

The vector is said to be the sum of the vectors. Therefore we can write

(Triangular law of vector addition)

From the diagram, if we represent the vector by p and the vector by q, the the vector is represented by p+q. The law that is called the** triangular law of vector addition. **The vector p+q is called the resultant of the vector p and q.

From the definition of a negative vector, it can be seen thet subtraction of a vector is the same as the addition of its negative vector.

Consider the diagram below.

P

a-b a+b

S -b Q b R

From the diagram above,

+()

=

=

To demonstrate the sum and difference of two vectors,

If = a, = b, then

= and = a-b

Note, in addition of two or more vectors, the x components will be added together and the y components will also be added together.

Example1. If vector and vector find a + b

Solution:

let and the

= ()

Example2. Add the following vectors and vector

Solution:

let and the

= (1+3, 2+5, 3+3, 4+7)

We can then add vectors by **adding the x parts** and **adding the y parts**

**example**

The vector (8, 13) and the vector (26, 7) add up to the vector (34, 20)

Subtraction of vectors:

**a** = **v** + −**k**

**a** = (12, 2) + −(4, 5)

= (12, 2) + (−4, −5)

= (12−4, 2−5)

= (8, −3)

Solution

(3, 3, 3, 3) + −(1, 2, 3, 4)

= (3, 3, 3, 3) + (−1,−2,−3,−4)

= (3−1, 3−2, 3−3, 3−4)

= (2, 1, 0, −1)

When we multiply a vector by a scalar it is called "scaling" a vector, because we change how big or small the vector is.

**a** = 3**m**

= (3×7, 3×3)

= (21, 9)

It still points in the same direction, but is 3 times longer

Magnitude of a Vector

The magnitude of a vector is shown by two vertical bars on either side of the vector:

**|a|**

OR it can be written with double vertical bars (so as not to confuse it with absolute value):

||**a**||

|**a**| = √(x^{2} + y^{2})

|**b**| = √ (6^{2} + 8^{2})

= √(36+64)

= √100

= 10

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