Further Mathematics

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.

                 

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:

Example1: subtract k = (4, 5) from v = (12, 2)

a = v + −k

a = (12, 2) + −(4, 5)

= (12, 2) + (−4, −5)

= (12−4, 2−5)

= (8, −3)

Example2: subtract (1, 2, 3, 4) from (3, 3, 3, 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)

Multiplying a Vector by a Scalar

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

Example: multiply the vector m = (7, 3) by the scalar 3

a = 3m 

= (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| = √(x² + y²)

Example: what is the magnitude of the vector b = (6, 8)?

|b| = √ (6² + 8²)

= √(36+64)

= √100

= 10

 

WEEK1

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