VECTOR AND SCALAR

LECTURENOTES

VECTORANDSCALAR

PREPAREDBY
WIDDEYKHALSOM BINTIEDRIS

Vector and Scalar

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WIDDEY KHALSOM BINTI EDRIS

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Vector and Scalar

All praise and gratitude to Allah for facilitating and allowing this e-book entitled Vector and Scalar to
be completed. I sincerely express my deep gratitude to En Hairman bin Omar, Head of Mathematics,
Science & Computer(JMSK) Department for the trust, support and opportunity for me to involve in
this project. Besides that, I would like to my give special appreciation to Puan Siti Fatimatuzahrah binti
Khairudin, Head of Mathematics Course, Puan Siti Noor Sarah binti Daud, Head of e-Learning Unit and
Puan Fazlina binti Yunus, lecturer of JMSK for all the support, encouragement and brilliant thought in
order for me to improve the writing of this lecture notes. Lastly, I hope that this lecture notes can be
beneficial to students and help them for a better understanding regarding this topic.

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Vector and Scalar

Vector and Scalar is one the chapter that is taught in Engineering Mathematics Course. This topic
explains the basic operations of vector and scalar quantities including their use in solving problems.
Besides that, this topic also explains the method for determining angle between two vectors as well
as the characteristics of triple vector and scalar products. The learning outcomes of this lecture notes,
students will be able to describe the difference between vector and scalar quantities, identify the
magnitude and direction of a vector, explain the effect of multiplying a vector quantity by a scalar,
describe how one-dimensional vector quantities are added or subtracted, explain the geometric
construction for the addition or subtraction of vectors in a plane and distinguish between a vector
equation and a scalar equation.

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Vector and Scalar

VECTOR AND SCALAR

NO TOPIC AND SUB TOPIC PAGE
5
1.1 Definition of Scalar and Vector 6
7
1.1.1 Basic of Vectors 10
10
1.1.2 Magnitude and Unit Vector Calculation
14
1.2 Operation of Vectors 16
16
1.2.1 Addition and subtraction of vectors using Triangle Construction
17
Method
19
1.2.2 Addition and subtraction of vectors using Parallelogram Method
20
1.3 Vectors in Cartesian Plane
23

1.3.1 Expressing vectors in form of + ( ) 24
24
1.3.2 Addition and subtraction of vectors in Cartesian Plane. 26
26
(Vectors in two dimensions) 26
29
1.3.3 Expressing vectors in form of + + ( ) 32
32
32
1.3.4 Addition and subtraction of vectors in Cartesian Plane 33
35
(Vectors in three dimensions) 35
39
1.3.5 Multiplication vectors with scalar in Cartesian Plane 40
41
1.3.6 Multiplication of vectors by Scalar
1.3.7 Writing resultant Equation by using Position Vector
1.4 Scalar (Dot) Product of Two Vectors
1.4.1 Define Scalar (Dot) Product
1.4.2 Basic properties of Scalar (Dot) Product
1.4.3 Scalar (Dot) Product – Cartesian Form
1.5 Vector (Cross) Product of Two Vectors
1.5.1 Define Vector(Cross) Product
1.5.2 Basic Properties of Vector (Cross) Product
1.5.3 Vector (Cross) Product in Cartesian Plane
1.6 Area of Parallelogram
1.6.1 Area of Parallelogram by Using Vector
SELF ASSESSMENT
ANSWER
REFERENCE

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Vector and Scalar

VECTOR AND SCALAR

1.1 Definition of Scalar and Vector

Scalar quantity is defined as the physical quantity with magnitude and no direction.
Some physical quantities can be described just by their numerical value (with their respective units)
without directions (they don’t have any direction).
Examples of Scalar Quantities
There are plenty of scalar quantity examples, some of the common examples are:

• Pressure
• Mass
• Speed
• Distance
• Time
• Area
• Volume
• Density
• Temperature

A vector quantity is defined as the physical quantity that has both direction as well as magnitude.
Geometrically, we can picture a vector as a directed line segment, whose length is the magnitude of
the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its
head.

head

tail

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Vector and Scalar

Examples of Vector Quantities
Vector quantity examples are many, some of them are given below:

• Velocity
• Linear momentum
• Acceleration
• Displacement
• Force

1.1.1 Basic of Vectors

1. Vector notation and representation
• Vector can be denoted as ⃗A, (bold A) , ̃ or a (bold a)
• Drawing and labelling vectors ⃗ ⃗ ⃗⃗ ⃗ means vector from point A to point B.

̃ = a B

A

⃗ ⃗ ⃗⃗ ⃗ = ̃ = a

• Vectors can be expressed or written in 2 ways

i) matrix or column vector

= ( ), example = ( 3 )
10

ii) engineering notation

= + , example = 3 + 10

2. Equality of vectors

• Two vectors are equal if only they have the same magnitude and direction
• Mathematically, we can say that two vectors say A and B are equal if they satisfy the

following conditions:

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Vector and Scalar

A = B (Vector A and B are equal), If and only if |A| = |B| (Equal magnitude) and
A ↑↑ B (Same direction)

= 2 unit = 2 unit

∴ =

3. Negative Vectors

• Negative vector is a vector that points in the direction opposite to the reference
positive direction.

• Vector that has same magnitude but different direction.

̃ = 2 unit − ̃ = 2 unit

1.1.2 Magnitude and Unit Vector Calculation

1. Magnitude Vector
• The magnitude of a vector is the length of the vector. The magnitude of the
vector A is denoted as | |
• Formula for the magnitude of vector A in two dimensional can be written as

| | = √ 2 + 2

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Vector and Scalar

Example 1 Exercise 1
Calculate the magnitude of vectors = (54) Calculate the magnitude of vectors = (−510)

Solution Ans: 11.18
= (45)
Exercise 2
| | = √42 + 52 Calculate the magnitude of vectors ⃗ ⃗ ⃗⃗ ⃗ = (−24)

= √16 + 25
= √41
= 6.4

Ans: 4.47

Example 2 Exercise 1
Calculate the magnitude of vectors Calculate the magnitude of vectors
̃ = −6 − 9 ̃ = 11 − 8

Solution Ans: 13.6

̃ = −6 − 9 Exercise 2
Calculate the magnitude of vectors
| ̃ | = √(−6)2 + (−9)2 ̃ = − + 2

= √36 + 81

= √117
= 10.82

Ans: 2.24

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Vector and Scalar

2. Unit Vector (û)

• A unit vector is a vector that has a magnitude of 1 unit. A unit vector is also known as
a direction vector. It is represented using a lowercase letter with a cap (‘^’) symbol
along and the formula for a unit vector is as below:

û = ̃ means unit vector in the direction of ̃ .
|̃ |

Example 1 Exercise 1

Calculate û in the direction of ̃ = 5 − 10 Calculate û in the direction of ̃ = 8 + 12

Solution

Step 1

| ̃ | = √52 + (−10)2 = 11.18

Step 2 Ans: 0.55 i+ 0.83j
̃
Exercise 2
û = | ̃ |
Calculate û in the direction of ̃ = 3 + 1
5 − 10
= 11.18 2 2
= 5 − 10

11.18 11.18

= 0.45 − 0.89

Example 2 Ans: 0.95 + 0.32
Calculate û in the direction of ⃗ = (−42) Exercise 1
Solution Calculate û in the direction of ⃗⃗ ⃗⃗ ⃗ = (152)

Step 1 9

| ⃗ | = √(−2)2 + (4)2 = 4.47

Step 2 = (−42)
⃗ 4.47

û = | ⃗ |

−2 Vector and Scalar
= (4.447) ∶ (00..9382)

4.47 Exercise 2
Calculate û in the direction of ̃ = (−43.5.6)
= (−00.8.495)

∶ (−00.7.683)

1.2 Operations of Vectors

Operation of vectors can be defined as the process of addition, subtraction and multiplication on
vectors. There are two methods that can be used to perform addition and subtraction on vectors.

• Using Triangle Construction Method or Triangle Law
• Using Parallelogram Method or Parallelogram Law

1.2.1 Addition and Subtraction of Vectors Using Triangle Construction Method (Triangle Law)

1. Addition of Vectors

The operation to add two or more vectors together to form a vector sum is known as the

addition of vectors. Two vectors can be added together to determine the result (or

resultant).

Example of Case 1

⃗⃗ ⃗⃗ ⃗⃗ + ⃗ ⃗⃗ ⃗⃗ ⃗ = ? C

A BB

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Vector and Scalar

C C
resultant
R
AB A B

Resultant Vector Equation can be written as : ⃗⃗ ⃗⃗ ⃗⃗ + ⃗ ⃗⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗

Example of Case 2 R
resultant
⃗⃗⃗ ⃗ ⃗⃗ + ⃗⃗⃗ ⃗⃗ ⃗ = ?
R PQ
MOVE ⃗ ⃗ ⃗ ⃗ to Point Q

P QP Q

Resultant Vector Equation can be written as : ⃗⃗⃗ ⃗ ⃗⃗ + ⃗ ⃗⃗ ⃗⃗ ⃗ = ⃗⃗⃗ ⃗⃗ ⃗

Properties of Addition of Vectors

The addition of vectors differs from the addition of algebraic numbers. Here are some of the most
significant properties to think about when adding vectors:
1. Vector addition is commutative:

It means the order of vectors does not affect the result of the addition. For example, If two
vectors A and vector B are added together, then vector A + vector B is equal to vector B +
vector A

A+B=B+A

2. Vector addition is associative:
The mutual grouping of vectors has no effect on the result when adding three or more vectors
together.

(A + B) + C = A + (B + C)

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Vector and Scalar

3. Vector addition is distributive:

It indicates that the sum of scalar times the sum of two vectors equals the sum of the scalar
times of the two vectors separately. ( k is a scalar)

k(A + B ) = kA + kB

4. Existence of Identity:
For any vector A,

A+0=A

Here, 0 is the additive identity. ( 0 is vector zero)
5. Existence of inverse:

For any vector A,
A+(– A) = 0

So, an additive inverse exists for every vector.

Example 1 Exercise 1
Draw the resultant ( ̃) below. Draw the resultant ( ̃) below.

̃ ̃ ̃
Solution ̃
̃
̃

̃

12

Example 2 Vector and Scalar
Draw the resultant ( ̃) below.
Exercise 1
̃ Draw the resultant ( ̃) below.
̃
̃

̃

Solution

Move ̃ Move ̃

̃ or

̃

̃ ̃

2. Subtraction of Vectors
We know that two vectors, A and B, can be added together using vector addition, and the
resultant vector can be written as R = A + B. Similarly, if we want the subtraction of two vectors,
A and B is expressed mathematically as: R = A – B , alternatively as: R = A +(- B)

Thus, subtracting the two vectors is the same as adding vector A and vector B’s negative (i.e.,
B). The vectors B and –B will have the same magnitude, but -B’s direction will be opposite to
that of vector B.

Let’s consider the diagram below:

̃ − ̃ = ̃ + (− ̃ ) = ?

- ̃ ̃ ̃
̃

̃ ̃ ̃

Resultant Vector Equation can be written as : ̃ − ̃ = ̃ + (− ̃ ) = ̃

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Vector and Scalar

1.2.2 Addition and Subtraction of Vectors Using Parallelogram Method (Parallelogram Law)

Parallelogram law of vector addition states that If two vectors act along two adjacent sides of
a parallelogram (with magnitude equal to the length of the sides) both pointing away from
the common vertex, the resultant is represented by the diagonal of the parallelogram passing
through the same common vertex.

Let’s consider the diagram below:
If the starting point of two vectors ̃ and ̃ coincide as shown below, then the diagonal of
the parallelogram formed by these vectors represents the sum of these two vectors ̃ +
̃ ( ).

̃ ̃ + ̃
̃
1. Addition of Vectors
⃗ ⃗ ⃗⃗ ⃗ + ⃗ ⃗⃗ ⃗⃗ = ? RS
R

P QP Q

Resultant vector Equation can be written as : ⃗ ⃗⃗ ⃗ ⃗⃗ + ⃗ ⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗

2. Subtraction of Vectors

- ̃ ̃ ̃ ̃

̃ ̃ ̃

Resultant vector Equation can be written as : ̃ + ̃ = ̃

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Vector and Scalar

Example 1 Exercise 1
Find the resultant vector for the operation
vectors below : Determine the resultant vector for the
a. ⃗ ⃗ ⃗⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗
b. ⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗ operation vectors below :
c. ⃗ ⃗ ⃗⃗ ⃗ − ⃗ ⃗ ⃗⃗ ⃗
d. ⃗⃗ ⃗⃗ ⃗ − ⃗⃗ ⃗⃗ ⃗ a. ⃗ ⃗ ⃗⃗ ⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗

Solution Ans: ⃗⃗ ⃗⃗ ⃗

a. ⃗⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ⃗ = ⃗⃗ ⃗ ⃗⃗ ⃗ b. ⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗⃗ ⃗
b. ⃗⃗ ⃗⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗
c. ⃗⃗ ⃗⃗ ⃗⃗ − ⃗⃗⃗ ⃗⃗ ⃗⃗ = ⃗ ⃗ ⃗⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗⃗ ⃗ Ans: ⃗ ⃗ ⃗⃗ ⃗
d. ⃗⃗ ⃗⃗ ⃗ − ⃗⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗ ⃗⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗ +
c. ⃗⃗ ⃗⃗ ⃗ − ⃗⃗ ⃗⃗ ⃗
⃗ ⃗ ⃗⃗ ⃗⃗ = ⃗⃗ ⃗ ⃗⃗ ⃗
Example 2 Ans : ⃗ ⃗⃗ ⃗⃗ ⃗
Given PQRS is a parallelogram of vectors. ⃗ ⃗ ⃗⃗ ⃗ =
̃ and ⃗⃗ ⃗⃗ ⃗ = ̃ d. ⃗ ⃗ ⃗⃗ ⃗ − ⃗⃗ ⃗ ⃗⃗ ⃗

PQ Ans : ⃗ ⃗⃗ ⃗⃗ ⃗

Exercise 1
The diagram below shows ABCDE as a vector
triangle. BCDE is a straight line with ⃗⃗ ⃗⃗ ⃗ =
⃗ ⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗ . If ⃗ ⃗ ⃗⃗ ⃗ = 3 ̃ + 2 ̃ and ⃗ ⃗ ⃗⃗ ⃗
= ̃ − ̃ , express the vectors below in terms
of ̃ and ̃ :

DE

SR D

Express the vectors below in terms of ̃ and ̃ : ED C
C

a. ⃗ ⃗ ⃗⃗ ⃗⃗ ) ⃗⃗ ⃗ ⃗ A B

b. ⃗ ⃗ ⃗⃗ e) ⃗⃗ ⃗⃗ ⃗ a. ⃗ ⃗ ⃗⃗ ⃗

c. ⃗⃗⃗ ⃗⃗ f ) ⃗⃗ ⃗⃗ ⃗

Solution Ans: −3 ̃ - 2 ̃
a. ⃗⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗ = ̃
b. ⃗⃗ ⃗⃗ = ⃗ ⃗⃗ ⃗⃗ ⃗ ⃗ = ̃ b. ⃗ ⃗ ⃗⃗ ⃗
c. ⃗⃗⃗ ⃗⃗ = − ⃗⃗ ⃗⃗ ⃗ = − ̃
d. ⃗⃗ ⃗ ⃗ = −⃗ ⃗ ⃗⃗ ⃗⃗ = − ̃
e. ⃗⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ + ⃗⃗ ⃗ ⃗⃗ ⃗ = ̃ + ̃
f. ⃗⃗ ⃗⃗ ⃗ = −( ̃ + ̃ ) = − ̃ − ̃

Ans: 4 ̃ +̃
15

c. ⃗ ⃗ ⃗⃗ ⃗ Vector and Scalar
d. ⃗ ⃗ ⃗⃗ ⃗ Ans: 5 ̃

Ans: ̃ - 6 ̃

1.3 Vector in Cartesian Plane

The natural way to describe the position of any point is to use Cartesian coordinates.


1.3.1 Expressing Vectors in form of + ( ) (Vectors in two dimensions)

In two dimensions, we have a diagram like this, with an x-axis and a y-axis, and an origin O.
To include vectors in this diagram, we have a vector i associated with the x-axis and a
vector j associated with the y-axis.

j ⃗⃗⃗ ⃗⃗ ⃗ = 4i and ⃗⃗ ⃗⃗ ⃗ = 3j

R ⃗ ⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗

4i +3j

3j = 4i + 3j or (34)

O 4i Q i

IMPORTANT KEY POINT

In two dimensions, the unit vectors in the directions of the two coordinate axes are written

as i and j . If a point P has coordinates (x, y) then the position vector ⃗⃗ ⃗⃗ ⃗ may be written as a

combination of these unit vectors, ⃗ ⃗ ⃗⃗ ⃗ = xi + yj , or equivalently as a column vector

⃗ ⃗ ⃗⃗ ⃗ =
( )

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Vector and Scalar

Example 1 Exercise 1
Draw a directed line segment to represent Draw a directed line segment to represent
each vector below. each vector below.
a. ̃ = (−51)
a. ̃ = (68) . ̃ = (−−46)
Solution b. ̃ = (43)

a.

c. ⃗ ⃗ ⃗⃗ ⃗ = (−53)

b.
d. ⃗⃗ ⃗⃗ ⃗ = (−−36)

1.3.2 Addition and Subtraction of Vectors in Cartesian Plane (For Vectors in Two Dimension)

Example 1 Exercise 1
Given ̃ = 3 + 2 , ̃ = −4 + , Given ̃ = 5 , ̃ = 2 , ̃ = −4 +
̃ = 6 − 9 ̃ = + 6 . Calculate 8 ̃ = 10 − . Calculate

a. ̃ + ̃ a. ̃ − ̃
b. ̃ − ̃
c. ( ̃ + ̃) − ̃
d. ( ̃ − ̃ ) + ( ̃ − ̃ )

Solution Ans: 5i - 2j
. ̃ +̃
b. ̃ + ̃ + ̃
= (3 + 2 ) + (−4 + )
= 3 + 2 − 4 +
= 3 − 4 + 2 +
= − + 3

Ans: 6i - 9j
17

. ̃ − ̃ c. ( ̃ + ̃ ) − ̃ Vector and Scalar
Ans: i + 6j
= (3 + 2 ) – (6 − 9 )
= 3 + 2 – 6 + 9
= 3 – 6 + 2 + 9
= – 3 + 11

c. ( ̃ + ̃) − ̃ d. ( ̃ − ̃ ) + ( ̃ − ̃ )

= (3 + 2 ) + (6 − 9 ) − (−4 + )
= 3 + 2 + 6 − 9 + 4 − )
= 3 + 6 + 4 + 2 − 9 −
= 13 − 8

d. ( ̃ − ̃ ) + ( ̃ − ̃) Ans: i + 5j

= [( + 6 ) − (−4 + )] + [(−4 + ) −
( 6 − 9 )]
= [ + 6 + 4 − ] + [−4 + − 6 + 9 ]
= (5 + 5 ) + (−10 + 10 )
= 5 + 5 − 10 + 10
= −5 + 15

Example 2 Exercise 1
Given ̃ = (−62) , ̃ = (−−42) , Given ̃ = (20) , ̃ = (−−52) ̃ = (−101).
̃ = (−74) ̃ = (81). Calculate, Calculate,

a. ̃ + ̃ a. ̃ + ̃
b. ̃ + ̃ − ̃
c. ( ̃ − ̃ ) + ̃ b. ̃ + ̃ − ̃ Ans: (110)
d. ( ̃ − ̃) + ( ̃ − ̃)

Solution
a. ̃ + ̃

= (−62) + (−−42) = (−62−−24) = (−46)

b. ̃ + ̃ − ̃

Ans: (−55)

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Vector and Scalar

= (−−42) + (−74) − (18) = (−−24 + 7 − 81) c. ( ̃ − ̃ ) + ̃
− 4 −
= (−214)

c. ( ̃ − ̃ ) + ̃

= [(−74) − (−62)] + (18)
= [(−74+−26)] + (81)
= (−910) + (81) = (−910++18) = (−102)

d. ( ̃ − ̃ ) + ( ̃ − ̃) Ans: (−115)
d. ( ̃ + ̃ − ̃ ) − ( ̃ + ̃)
= [(−−42) − (−74)] + [(−62) − (18)]
Ans: (−−56)
= (−−42 − 47) + (−62−−81)
+

= (−211) + (−−32)

= (−211−−23)

= (−014)


1.3.3 Expressing Vectors in form of + + ( ) (Vectors in Three Dimensions)



In three dimensions we have three axes, traditionally labelled as x, y and z, all at right angles

to each other. Any point P can now be described by three numbers, the coordinates with

respect to the three axes.

Now let’s take a point P in three-dimensional space, with coordinates (2, 3, 5). The position
vector of the point will be the line segment ⃗⃗ ⃗⃗ ⃗ . We can now write ⃗⃗ ⃗⃗ ⃗ = 2 + 3 +
5 where k is a unit vector in the direction of the z-axis. Again it is important to appreciate

19

Vector and Scalar

the difference. The numbers (2, 3, 5) represent a set of coordinates, referring to the point P.
But the expression 2 + 3 + 5 is a vector, the position vector ⃗⃗ ⃗⃗ ⃗ . We sometimes write

2
this is as a column vector (3) means the same as 2 + 3 + 5

5

IMPORTANT KEY POINT

In three dimensions, the unit vectors in the directions of the three coordinate axes are written
as i, j and k. If a point P has coordinates (x, y, z) then the position vector ⃗⃗ ⃗⃗ ⃗ may be written
as a combination of these unit vectors, + + , or equivalently as a column vector


⃗ ⃗ ⃗⃗ ⃗ = ( )



1.3.4 Addition and Subtraction of Vectors in Cartesian Plane (For Vectors in Three Dimensions)

Example 1 Exercise 1
Given ̃ = 8 + 2 − 6 , ̃ = −3 + Given ̃ = 5 − , ̃ = 2 + 5 , ̃ =
, ̃ = 6 + + ̃ = −6 + −4 + 8 + 9 ̃ = 10 − 6 . Calculate
10 . Calculate
a. ̃ - ̃
a. ̃ + ̃ + ̃
b. ̃ − ̃
c. ( ̃ + ̃) − ̃
d. ( ̃ − ̃ ) + ( ̃ − ̃ )

Ans : 5i -3j -5k

Solution b. ̃ + ̃ + ̃
. ̃ + ̃ + ̃

= 8 + 2 − 6 + (−3 + ) + (6 + + )

= 8 + 2 − 6 − 3 + + 6 + +

= 8 − 3 + 6 + 2 + − 6 + +

= 11 + 3 − 4

. ̃ − ̃ Ans : 6i + 4j + 14k
= 8 + 2 − 6 − ( −3 + )
= 8 + 2 − 6 + 3 − c. ( ̃ + ̃ ) − ̃

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Vector and Scalar

= 8 + 3 + 2 − 6 −
= 11 + 2 − 7

Ans : I + 5j + 4k

c. ( ̃ + ̃) − ̃ d. ( ̃ − ̃ ) + ( ̃ − ̃ )
= (8 + 2 − 6 ) + (6 + + ) −
(−3 + )
= 8 + 2 − 6 + 6 + + + 3 −
= 8 + 6 + 3 + 2 + − 6 + −
= 17 + 3 − 6
d. ( ̃ − ̃ ) + ( ̃ − ̃)
= [(−6 + 10 ) − (−3 + )] + (−3 +
) − ( 6 + + )]
= −6 + 10 + 3 − − 3 + −
6 − −
= 3 − 3 − 6 − 6 − + 10 − +
−
= − 6 − 7 + 9

Ans : i + j + 4k

Example 2 Exercise 1

−2 −4 0 −5 10
Given ̃ = (5) , ̃ = (−2) ̃ = ( 1 ).
Given ̃ = ( 6 ) , ̃ = ( 1 ) ,

3 −2 Calculate, 26 8
−4 1
̃ = ( 7 ) ̃ = (0). Calculate,
58
a. ̃ + ̃

a. ̃ + ̃ 10
Ans : ( 6 )
b. ̃ + ̃ − ̃
c. ( ̃ − ̃ ) + ̃ 10
d. ( ̃ − ̃) + ( ̃ − ̃)

21

Vector and Scalar

b. ̃ + ̃ − ̃

Solution
a. ̃ + ̃

−2 −4 −2 − 4 −6
= ( 6 ) + ( 1 ) = ( 6+1 ) = ( 7 )

3 −2 3 − 2 1

b. ̃ + ̃ − ̃ 5
Ans : (−6)
−4 −4 1 −4 − 4 − 1 c. ( ̃ − ̃) + ̃
12

= ( 1 ) + ( 7 ) − (0) = ( 1 + 7 − 0 )
−2 5 8 −2 + 5 − 8
−9
= (8)

−5

c. ( ̃ − ̃ ) + ̃

−4 −2 1
= [( 7 ) − ( 6 )] + (0)
538

−4 + 2 1 −15
= [( 7 − 6 )] + (0) Ans : ( 2 )

5−3 8 0
−2 1 −2 + 1 1 d. ( ̃ + ̃ − ̃ ) − ( ̃ + ̃)
= ( 1 ) + (0) = ( 1 + 0 ) = ( 1 )

2 8 2+8 10

d. ( ̃ − ̃) + ( ̃ − ̃)

−4 −4 −2 1
= [( 1 ) − ( 7 )] + [( 6 ) − (0)]
−2 5 38

−4 + 4 −2 − 1 −5
=( 1−7 ) + ( 6−0 ) Ans : (−12)

−2 − 5 3−8 2

0 −3
= (−6) + ( 6 )

−7 −5

0−3
= (−6 + 6)

−7 − 5

−3
=( 0 )

−12

22

Vector and Scalar

1.3.5 Multiplication Vectors With Scalar in Cartesian Plane
If ( ) = ( )
Consider m is a scalar, therefore, ( + ) = + and

Example 1 Exercise 1
Given ̃ = 2 + 3 , ̃ = 4 − 5
Given ̃ = −4 + 8 , find
̃ = −5 + 9 . Find,
. 3 ̃ . 1 ̃ a. 3 ̃ + 2 ̃ − ̃
2

Solution

a. 3 ̃
= 3(−4 + 8 )
= −12 + 24

1 Ans: 19i - 10j
b. 2 ̃
1 b. | ̃ − 2 ̃|
2
= (−4 + 8 )

= −2 + 4

Example 2 Ans: 26.93
Exercise 1
Given ̃ = (−92), find b. 2 ̃ Giveñ = (−71) , ̃ = (−64) ̃ = (38),
a. 6 ̃ find
3 a. ̃ − 2 ̃ + 4 ̃

Solution ∶ (1297)
b. ℎ ℎ
a. 6 ̃
2 ̃ + ̃
= 6 (−92) = (−5142)

b. 2 ̃ = 2 (−92) = 6
3 (− 4)
3
3

13

Ans :(2105)

20

23

Vector and Scalar

1.3.6 Multiplication of Vectors by Scalar
If ̃ multiply by a scalar, k, the product is a vector denoted as k ̃ .
k ̃ means a vector that parallel with ̃ and has k times magnitude ̃
̃

2 ̃

1

̃

2

IMPORTANT KEY POINTS
If is positive, then the vector ̃ ̃ are parallel and have the same direction.
If is negative, then vector ̃ ̃ are parallel but have opposite direction.

Example 1 Exercise 1
Determine
Calculate 4⃗ ⃗ ⃗⃗ ⃗ + 9 ⃗⃗ ⃗⃗ ⃗ + 3 ⃗⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗ + 8 ⃗⃗ ⃗⃗ ⃗
a. 6⃗ ⃗ ⃗⃗ ⃗ + 5 ⃗⃗ ⃗⃗ ⃗
b. 7⃗ ⃗ ⃗⃗ ⃗ − 5 ⃗⃗ ⃗⃗ ⃗
c. 4⃗ ⃗ ⃗⃗ ⃗ + 2⃗ ⃗ ⃗⃗ ⃗
d. 3 ⃗⃗ ⃗⃗ ⃗ + 3⃗ ⃗ ⃗⃗ ⃗

Solution 6⃗ ⃗ ⃗⃗ ⃗ + 5 ⃗⃗ ⃗⃗ ⃗ = 11⃗ ⃗ ⃗⃗ ⃗
a. 7 ⃗⃗ ⃗⃗ ⃗ − 5⃗ ⃗ ⃗⃗ ⃗ = 2 ⃗⃗ ⃗⃗ ⃗
b. 4⃗ ⃗ ⃗⃗ ⃗ + 2 ⃗⃗ ⃗⃗ ⃗ = 4 ⃗⃗ ⃗⃗ ⃗ − 2⃗ ⃗ ⃗⃗ ⃗ = 2 ⃗⃗ ⃗⃗ ⃗
c. 3⃗ ⃗ ⃗⃗ ⃗ + 3⃗ ⃗ ⃗⃗ ⃗ = 3(⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗ ) = 3⃗ ⃗ ⃗⃗ ⃗
d.

Ans: 9 ⃗⃗ ⃗⃗ ⃗

1.3.7 Writing Resultant Equation By Using Position Vector

Position vector is defined as vector that which relative to origin. Example, ⃗⃗ ⃗⃗ ⃗ , ⃗⃗ ⃗⃗ ⃗ , ⃗ ⃗ ⃗⃗ ⃗ ,

Resultant Vector Equation can be written as,

⃗ ⃗ ⃗⃗ ⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗
⃗⃗ ⃗⃗ ⃗ = ⃗⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗
⃗ ⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗ ⃗ + ⃗ ⃗⃗ ⃗⃗ ⃗

24

Vector and Scalar

IMPORTANT KEY POINT
In the following diagram, point A has the position vector ̃ and point B has the position
vector ̃ .
If ̃ = (25), the coordinates of A will be (2,5)
Similarly, If ̃ = (43), the coordinates of B will be (4,3)
y
A (2,5)
B (4,3)
x

Example 1 Exercise
Given four points as A, B, C and D which have By using the same example above, calculate,
position vectors as a. ⃗⃗ ⃗⃗ ⃗
⃗⃗ ⃗⃗ ⃗ = ̃ , ⃗⃗ ⃗⃗ ⃗⃗ ⃗ ⃗ = ̃ , ⃗ ⃗ ⃗⃗ ⃗ =̃ + 2 ̃ and
⃗ ⃗⃗ ⃗⃗ ⃗ = 2 ̃ − ̃ . Express the operation of Ans: ̃ + ̃
vectors below in terms of ̃ and ̃ .
b. ⃗ ⃗ ⃗⃗ ⃗
Solution

⃗⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗⃗ ⃗
= − ⃗ ⃗ ⃗⃗ ⃗⃗ + ⃗⃗ ⃗⃗ ⃗

= − ̃ + ̃
= ̃ − ̃

c. ⃗⃗ ⃗⃗ ⃗ Ans: ̃ − 3 ̃
d. ⃗ ⃗⃗ ⃗⃗ ⃗ Ans: 2 ̃

: 2 ̃ − 2 ̃
25

Vector and Scalar
1.4 Scalar (Dot) Product of Two Vectors

The Scalar product is also known as the Dot product, and it is calculated in the same manner
as an algebraic operation. In a scalar product, as the name suggests, a scalar quantity is
produced. (not a vector)

1.4.1 Define Scalar (Dot) Product

Scalar Product also known as Dot Product can be defined as

̃ ● ̃ = | ̃ || ̃ | cos Ɵ
Where:
| ̃ | is the magnitude (length) of vector ̃
| ̃ | is the magnitude (length) of vector ̃
θ is the angle between ̃ and ̃

B ⃗ ⃗ ⃗⃗ ⃗ = ̃ ⃗⃗ ⃗⃗ ⃗ = ̃

 Ɵ is angle between ⃗ ⃗ ⃗⃗ ⃗ ⃗ ⃗ ⃗⃗ ⃗

O A

1.4.2 Basic Properties of Scalar (Dot) Product
) ̃ ● ̃ = ̃ ● ̃
) ̃ ● ̃
) ( ̃ ● ̃ ) ● ̃ ( ̃ ● ̃ ) ̃

d) Parallel directions of two vectors ̃

̃
and

If ̃ and ̃ are parallel and in the same directions, so  = 0

26

Vector and Scalar

Therefore,
̃ ● ̃ = | ̃ || ̃ | cos 0° = | ̃ || ̃ | (1) = | ̃ || ̃ |

e) Opposite directions of two vectors ̃
̃
and

If ̃ and ̃ are parallel and in opposite directions, so  = 180

Therefore,
̃ ● ̃ = | ̃ | | ̃ | cos 180° = | ̃ | | ̃ | (−1) = −| ̃ | | ̃ |

f) Perpendicular vector

If ̃ and ̃ are perpendicular to each other, so  = 90
Therefore,
̃ ● ̃ = | ̃ || ̃ | cos 90° = | ̃ | | ̃ | (0) = 0

g. An angle between two vectors

 cos = ̃ ●̃
| ̃ | | ̃ |

27

Vector and Scalar

Example 1 Exercise 1
Given | ̃ | = 3.5, | ̃ | = 4.2 and angle
between ̃ ̃ is 75°. Based on the diagram below, |⃗ ⃗ ⃗⃗ ⃗ | =
Determine ̃ ● ̃ 5 , | ⃗⃗ ⃗⃗ ⃗ | = 6 | ⃗⃗ ⃗⃗ ⃗ | = 4.
Solution
C
̃ ● ̃ = | ̌| | ̃ | cos 75°
= (3.5) (4.2) cos 75° 40 B
= 14.7 cos 75° O 20 A
= 3.8
Calculate,
a. ⃗ ⃗ ⃗⃗ ⃗ ● ⃗⃗ ⃗⃗ ⃗

Ans: 28.19

b. ⃗ ⃗ ⃗⃗ ⃗ ● ⃗⃗ ⃗⃗ ⃗

Ans: 18.39

c. ⃗ ⃗ ⃗⃗ ⃗⃗ ●⃗⃗ ⃗⃗ ⃗ ⃗

Ans: 10

d. ⃗ ⃗ ⃗⃗ ⃗ ● ⃗ ⃗ ⃗⃗ ⃗

Ans: 25
28

Vector and Scalar

Example 2 Exercise 1

If ⃗ ● ⃗ = 2, | ⃗ | = 3 | ⃗ | = 4 , find Determine the angle between ̃ ̃ if

angle between ⃗ ⃗ . | ̃ | = 4 , | ̃ | = 5 ̃ ● ̃ = 6

Solution

Cos  = a~ • b~
a~ b~

= 2
(3)(4)

=0.1667

= 80.41°

1.4.3 Scalar (Dot) Product in Cartesian Form Ans: 72.54°
29
Cartesian form expressed as +

Consider ,

~r1 = 1 + 1 or ( x1 )
y1

~r2 = 2 + 2 or ( x2 )
y2

Therefore,

~r1 • ~r2 = x1 x2 + y1 y2
If  is angle between ~r1 and ~r2 , therefore,



Vector and Scalar

cos  = ~r1 • ~r2
~r1 ~r2

Example 1 Exercise 1

Given ~r1 = 2 + and ~r2 = − 3 , Given ~r1 = 5 + 4 and ~r2 = − 2 ,

calculate find,

~r1 ● ~r2 a. ̃ 1 ● ( ̃ 2 − ̃ 1)

Solution

~r1 = 2 + = (21) and ~r2 = − 3 =

(−13)

~r1 ● ~r2 = (21) ● (−13) Ans : -44

b. (3̃ 1 ● ̃ 2) ̃ 1

= (2)(1) + (1)(−3)

= −1

Ans : -45i-36j

c. (10 ̃ 1 + ̃ 2) ● ̃ 1

Ans: 407

d. Angle between ̃ 1 ̃ 2

Ans: 102.1°
30

Vector and Scalar

Cartesian form expressed as + +

Consider ,

~r1 = 1 + 1 + 1 x1
or ( y1 )



~r2 = 2 + 2 + 2 x2
or ( y 2 )



~r1 • ~r2 = 1 2 + 1 2 + 1 2

If  is an angle between ~r1 and ~r2 , therefore,



cos  = ~r1 • ~r2
~r1 ~r2

Example 1 Exercise 1

~r1 = 2 + 3 − and ~r2 = 3 + . ~r1 = 8 + − 2 and ~r2 = 4 + , find,

Calculate . 2 ̃ 1 ● 3 ̃ 2

~r1 • ~r2

Solution

~r1 = 2 + 3 − = 2
(3)

−1 Ans : 180

~r2 3 b. ( ̃ 1 + ̃ 2) ● (̃ 2 − ̃ 1)
= (0)
= 3 +

1

~r1 • ~r2 = 2 3
( 3 )• (0)

−1 1

Ans : -52
31

Vector and Scalar

= (2)(3) + (3)(0) + c. Angle between ̃ 1 ̃ 2
(−1)(1)

=6 + 0 − 1 = 5

Ans : 28.89°
d. Angle between 2 ̃ 1 3 ̃ 2

Ans :
28.82 °

1.5 Vector (Cross) Product of Two Vectors
1.5.1 Define Vector (Cross) Product

Vector (Cross) Product can be defined as

̃ × ̃ = | ̃ || ̃ | sin . ̂

| ̃ | = magnitude ̃
| ̃ | = magnitude ̃
= angle between ̃ and ̃
̂ = unit vector that in the direction of ̃ × ̃
̃ × ̃

̃ ̃
̂



O

1.5.2 Basic Properties Of Vector (Cross ) Products.
1. ̃ ● ( ̃ × ̃ ) = ( ̃ × ̃ ) ● ̃
2. ̃ × ̃ = − ̃ × ̃ * Note that c is a scalar
3. ( ̃ ) × ̃ = ( ̃ × ̃ ) = ̃ × ( ̃ )
4. ̃ × ( ̃ + ̃) = ̃ × ̃ + ̃ × ̃
5. | ̃ × ̃ | = | ̃ | | ̃ | sin
6. An angle between ̃ and ̃

| ̃ × ̃ |
sin = | ̃ | | ̃ |

32

Vector and Scalar

7. If ̃ is parallel with ̃  = 00 or 1800

| ̃ × ̃ | = | ̃ | | ̃ | sin 0

= | ̃ | | ̃ |(0)

=0  = 900
8. If ̃ is perpendicular to ̃

| ̃ × ̃ | = | ̃ || ̃ | 900

= | ̃ | | ̃ |(1)

= | ̃ | | ̃ |

1.5.3 Vector ( Cross ) Product in Cartesian Form.

Consider,

1

̃ = 1 + 1 + 1 or ( 1)
1
2
̃ = 2 + 2 + 2 or ( 2)
2


̃ × ̃ = | 1 1 1|

2 2 2
= | 12 1 | 12 1 | 12 12|
2 | − 2 | +

= ( 1 2 − 1 2) − ( 1 2 − 1 2) + ( 1 2 − 1 2)

Example 1: Exercise 1
Given that ⃗⃗⃗ = + 2 + 3 ⃗ = Given that ⃗⃗⃗ = 3 − 3 + , ⃗ = 4 +
9 + 2 . Calculate ⃗⃗⃗ × ⃗⃗ ⃗
4 + 5 + 6
Determine ̃ × ̃

Solution


̃ × ̃ = |1 2 3|
456

= |25 36| − |41 63| + |14 52|
= ((2)(6) − (3)(5)) − ((1)(6) −

(3)(4)) + ((1)(5) −

(2)(4))

= (12 − 15) − ( 6 − 12)
+ (5 − 8)

= −3 + 6 − 3

Ans : −15i – 2 + 39k

33

Vector and Scalar

Example 2: Exercise 1
3
3
Determine vector product of ̃ = ( 2 ) and Given that ̃ = (−2) ̃ =
−2
−2
1 −1
̃ = ( 0 ) ( 0 ) Calculate ̃ × ̃
5
−5
Solution


̃ × ̃ = |3 2 −2|

1 0 −5
= |02 −−25| − |13 −−25| + |31 20|
= ((2)(−5) − (0)(−2))

− ((3)(−5) − (1)(−2))

+ ((3)(0) − (1)(2))

= (−10) − ( − 15 + 2)

+ (0 − 2)

= −10 + 13 − 2

Example 2 Ans : −10i – 13 - 2k
Given that ̃ = + 3 , ̃ = 2 − + Exercise 1
3 . Calculate ( ̃ × ̃ ) • ̃ Given that ̃ = 5 − 2 + 3 , ̃ = 3 +
2 and
Solution ̃ = −3 + 4 . Determine,

a. ( ̃ × ̃ ) X ̃
̃ × ̃ = |1 0 3|
2 −1 3

= |−01 33| − |21 33| + |21 −01|

= ((0)(3) − (3)(−1))
− ((1)(3) − (3)(2))
+ ((1)(−1) − (0)(2))

= (0 − (−3) − (3 − 6)
+ (−1 − 0)

= 3 + 3 −

: 14 + 16 + 12
34

Vector and Scalar

32 b. 2 ̃ ●( ̃ × ̃ )
( ̃ × ̃ ) • ̃ = ( 3 ) ● (−1)

−1 3

= (3)(2) + (3)(−1) + (−1)(3)
=6−3−3
=0

Ans: 54
c. Angle between 2 ̃ ( ̃ × ̃ )

1.6 Area of Parallelogram Ans: 74.27°
1.6.1 Area of Parallelogram by Using Vector 35

t




⃗

Vector and Scalar

Consider ⃗⃗ ⃗ and ⃗ are side by side, therefore,

Area of parallelogram = | || ⃗ | sin

= | × ⃗ |

Area of triangle = × | × ⃗ |


Example 1 Exercise 1
Find the area of the parallelogram vector if Find the area of the parallelogram spanned by
vector ⃗ ⃗ ⃗⃗ ⃗ = − − 5 + 2 and vector ⃗⃗ ⃗⃗ ⃗ = + 2 + 3 and ⃗ ⃗ ⃗⃗ ⃗ = 2 +
− 2
⃗⃗ ⃗⃗ ⃗ ⃗ = 2 + 3

Solution


⃗⃗ ⃗⃗ ⃗ × ⃗⃗ ⃗⃗ ⃗ = |−1 −5 2|

2 03

= |−05 32| − |−21 32|
+ |−21 −05|

= (−15 − 0) − (−3 − 4)
+ (0 + 10)

= −15 + 7 + 10

Area of parallelogram Ans : 11.045 2
=| ⃗⃗ ⃗⃗ ⃗ × ⃗⃗ ⃗⃗ ⃗ |
=√(−15)2 + 72 + 102
=√225 + 49 + 100
=√374
=19.34 2

36

Vector and Scalar

Example 2 Exercise 1
A, B and C is a triangle with vertices of
Given position vector of point A,B and C are A(3,0,2), B(4,6,1) and C (0,5,4). Calculate area
of triangle ABC.
2 −3 5
⃗ ⃗ ⃗⃗ ⃗ =(6) , ⃗ ⃗ ⃗⃗ ⃗ = ( 1 ) and ⃗ ⃗ ⃗⃗ ⃗ =(4)

04 2

Calculate the area of parallelogram and
triangle for vectors ⃗ ⃗ ⃗⃗ ⃗ and ⃗ ⃗ ⃗⃗ ⃗ .

Solution

⃗⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗
= − ⃗ ⃗ ⃗⃗ ⃗ + ⃗ ⃗ ⃗⃗ ⃗

2 −3
= − (6) + ( 1 )

04

−5
= (−5)

4

⃗⃗ ⃗⃗ ⃗ = ⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗

= − ⃗ ⃗ ⃗⃗ ⃗ + ⃗⃗ ⃗⃗ ⃗

25
= − (6) + (4)

02

3
= (−2)

2


⃗ ⃗ ⃗⃗ ⃗ × ⃗⃗ ⃗⃗ ⃗ = |−5 −5 4|

3 −2 2

= |−−52 24| − |−35 24| + |−35 −−25|

= (−10 + 8) − (−10 − 12)
+ (10 + 15)

= −2 + 22 + 25

Area of parallelogram

=| ⃗⃗ ⃗⃗ ⃗ × ⃗ ⃗ ⃗⃗ ⃗ |

=√(−2)2 + 222 + 252

=√4 + 484 + 625 Ans : 14.31 2
37

Vector and Scalar

=√1113

=33.36 2

Area of triangle

= 1 × 33.36 = 16.68 2
2

38

Vector and Scalar

SELF ASSESSMENT

1. Given ̃ 1 = 2i + j and 2̃ = i – 3j , find
a. 1̃ • 2̃
b. 1̃ • ( 2̃ – 1̃ )
c. (10 1̃ + 2̃ ) • 1̃
62

2. Given ⃗⃗ ⃗⃗ ⃗ = (−3) and ⃗ ⃗ ⃗⃗ ⃗ = ( 4 ), find
1 −5

a. ⃗⃗ ⃗⃗ ⃗⃗ ● ⃗⃗ ⃗⃗ ⃗
b. ⃗⃗ ⃗⃗ ⃗⃗ × ⃗⃗ ⃗⃗ ⃗
c. ⃗ ⃗ ⃗⃗ ⃗

3. Position vector for point A, B and C are ̃ = 2 + 3 + , ̃ = 4 + 2 + 3
̃ = − 3 + 2 respectively. Find

a. ̃ + ̃
b. ̃ + ̃

c. ⃗⃗ ⃗⃗ ⃗
d. Unit vector in the direction of ̃ + ̃ − ̃

4. Given two vectors P(4,10) and Q(8,6).

a. Sketch vector ⃗ ⃗ ⃗⃗ ⃗ by using triangle methods

b. Determine the value of vector ⃗⃗ ⃗⃗ ⃗

c. Calculate the magnitude of vector ⃗⃗ ⃗⃗ ⃗

5. Prove that ̃ . ( ̃ × ̃) = ( ̃ × ̃ ) . ̃

1 −1 5

If given ̃ = (2) , ̃ = ( 0 ) and ̃ = (4)

43 2

39

Vector and Scalar

ANSWERS

1. a. -1 b. -6 c. 49

−4
2. a. -5 b. 11i + 32j + 30k c. ( 7 )

−6

3. a. 6i +5j+4k b. 5i-j+5k c. –i-6j+k 5i + 8 j + 2k

d.

93

4. a.

b. -4i + 4j
c. 5.657 units

5. 6 = 6 (Proven)

40

Vector and Scalar

REFERENCES

1. Bird, J. (2010). Higher Engineering Mathematics. Science & Technology Rights Department in
Oxford, UK. Published by Elsevier Ltd.

2. Bird, J.O & May, A.J.C. (1994). Technician Mathematics. Longman Group UK Limited.
3. Diana Malini Jarni, Norazila Mad & Arffaazila Rahmat. (2018). Analysis of Final Examination

Question, Jun 2018 Edition. Politeknik Sultan Salahuddin Abdul Aziz Shah, Selangor.
4. Frank, D & Nykamp, DQ. (n.d). An Introduction to Vectors. Retrieved from

https://mathinsight.org/vector_introduction
5. Nykamp, DQ. (n.d). Math Insight. Retrieved from

https://mathinsight.org/definition/magnitude_vector
6. Rozinah @Nurhaizi Ramli, Elisnorazmaliza Ab Hamid, Hartati Maskur, Hartini Hardono & Ira

Fazlin Mohd Fauzi. (2015). Engineering Mathematics 1, Guide To Problem Solving. (Third
Edition) . Politeknik Kota Bharu, Kelantan.
7. Zuraidah Mohd Ramly, Norihan Mahmood, Suhana Abdul Aziz & Roveena Harleen Hussain
Meah. (2017).Engineering Mathematics Vol 1.Department of Mathematics, Science &
Computer, Politeknik Ibrahim Sultan, Johor.

41