Engineering Mathematics 1

[CHAPTER 5: VECTOR AND SCALAR]

O, A, B,C and D are five points where OA  a , OB  b , OC  a  2b and OD  2a  b .
Express AB, BC,CD, AC and BD in terms of a and b.

Ans:

AB  b  a
BC  a  b
CD  a  3b
AC  2b
BD  2a  2b

LET’S PRACTICE 6

VWXY is a parallelogram with VW  a and WX  b . Express the vectors below in terms of a and
b.

i. VX Ans: a  b
ii. XV Ans:  b  a
iii. WY

Ans: b  a

iv. YW

Ans: a  b
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[CHAPTER 5: VECTOR AND SCALAR]

6.3 APPLY SCALAR (DOT) PRODUCT OF TWO VECTORS

SCALAR (DOT) PRODUCT
 Define scalar product
 State properties of scalar product
 Calculate the scalar product

Definition of scalar product

 If the component vector of A  a1i  a2 j  a3k and B  b1i  b2 j  b3k

Then the scalar product can be defined by:

 
A  B  A B cos 

Properties of scalar product

i.  
AB  B A

ii. For non zero vector A and B ;
BA0 

Ifandonlyif Ais perpendicular to B
A BC  AB AC
 iii.

    
A kB  kA  B  k A B
     iv.

v. If vectors  and  are given in term of their component with respect to the standard
A B

vectors i, j and k as

 j
A  a1i  a2  a3k

B  b1i  b2 j  b3k

Then;


A B  a1b1  a2b2  a3b3

149

[CHAPTER 5: VECTOR AND SCALAR]

EXAMPLE 6

1. Given that OQ  6i  3 j  k and OR  2i  4 j 5k , find the scalar product for the

vectors.

SOLUTION

OQ  OR  62  3 4 1 5

OQ  OR  19

LET’S PRACTICE 7


1. If P  2i  4 j  6k and U  2i  4 j  3k . Find

a.     
P U
b. P U P

2. Given A4,4,8, B5,1,8 and C 2,4,3. Find: Ans: 38 , 76i 152 j  228k

 Ans: 80
i. A  B Ans: 32
Ans: 10

ii. A  C 150

iii. 
BC

[CHAPTER 5: VECTOR AND SCALAR]

6.4 APPLY VECTOR (CROSS) PRODUCT OF TWO VECTORS

Properties of vector product

If A, B and C are vectors and d is a scalar, then

i. A  B  B  A

ii. dA B  d A  B  A  dB 
iii. A  B  C  A  B  A  C
iv. A  B C  A  C  B  C
v A B C  A B C
vi. A  B  C  A  CB  A  BC

EXAMPLE 7


1. Find the vector product for vector A  2i  2 j  2k and B  2i  2 j  3k . Find

SOLUTION:

i j k 
A  B  2 2  2

2  2 3 

 23  2 2i  23  22j  2 2 22k

 2i 10 j  8k

2. Find the unit vector of u  v . Given u  2i  2 j  3k and v  i  3 j  k .

SOLUTION:

i j k 
u  v  2 2  3  11i  5 j  4k

1 3 1 

u  v  112   52  42  162

Vector unit of u  v is;  11 i  5 j 4k
162 162 162

151

[CHAPTER 5: VECTOR AND SCALAR]

LET PRACTICE 8

1. Given vector OA 2i  j  3k , OB  3i  2 j  4k and OC  i  3 j  2k . Determine

a. AB

 b. OA OBOC Ans: i  3 j  7k

 c. OAOB OC Ans: 39

 Ans:  55i 11 j 11k
2. If P 8i  5 j  4k and Q  2i  7 j  4k , find:
Ans:  48i  24 j  66k
a. Q  P
Ans: 48i  24 j  66k
 Ans:  8i 13 j  10k
b. P  Q
Ans: 8i  13 j  10k
3. If a  3i  2 j  5k and b  i  4 j  6k , find: 152
a. a  b
b. b  a

[CHAPTER 5: VECTOR AND SCALAR]
6.4.1 APPLICATION OF THE VECTOR (CROSS) PRODUCT

Area of parallelogram

 AB AC sin
 AB BC

Area of triangle ABC

 1 AB BC
2

EXAMPLE 8

1. Find the area of the parallelogram with vertices A0,5, B2,0 ,  8,1 and D 6,4.
C

SOLUTION:

AB  2,5  2i  5 j  0k

BC  6,1  6i  j  0k

Area of parallelogram Area  AB  BC

i jk
AB  BC  2  5 0

6 1 0

AB  BC  0i  0 j  28k
AB  BC  02  02  282  28

153

[CHAPTER 5: VECTOR AND SCALAR]

2. Calculate the area of the parallelogram spanned by the vectors   3
P  3

 1 

and   4
Q 9

2

SOLUTION:

P  3,3,1  3i  3 j  k

Q  4,9,2  4i  9 j  2k

Area of parallelogram Area  P Q

i jk
PQ  3 3 1

492

P  Q   6  9i  6  4 j  27 12k

P  Q  15i  2 j  39k

P  Q  152   22  392  41.83

LET PRACTICE 9

1. Find the area of parallelogram with vertices A0,0, B2,1 , C3,6 and D 1,5.

Ans: 9

154

[CHAPTER 5: VECTOR AND SCALAR]
2. Find the area of parallelogram with U  i  j  3k and V  6 j  5k .

Ans: 230

3. Find the area of triangle with vertices P1,1,0, Q 2,0,1 and R0,2,3.

Ans: 4.899

4. Find the area of parallelogram with vertices P1,5,3, Q0,0,0 and R3,5,1.

Ans: 24.5

155

[CHAPTER 5: VECTOR AND SCALAR]

5. Find the area of parallelogram with vertices x2,0,3, y1,4,5, and z7,2,9.

Ans: 64.9

156

TOPICS INSIDE

1.0 BASIC ALGEBRA

This topic to introduces basic algebraic concept and its use in solving or
simplifying linear, polynomials and quadratic equations.

2.0 PARTIAL FRACTION

This topics discusses about proper and improper fractions including partial fraction
decomposition

3.0 TRIGONOMETRY

This topic discusses about proper and improper fractions including partial fraction
decomposition.

4.0 COMPLEX NUMBER

This topic explains the fundamental concept of trigonometric functions particularly
on the six trigonometric ratios of special angles and simple trigonometric basic
identities. The topic also explains about trigonometric identities, sine and cosine
rules. Skills using trigonometric identities, sine cosine rules to solve simple
trigonometric equations are discussed.

5.0 MATRICES

This topics introduces the type and characteristics of matrix up to 3x3 matrix. This
topic also explain the operation involving matrices such as addition,subtraction
and multiplication of matrices. The inverse matrix method and Cramer’s Rule is
also explain to solve simultaneous equation up to three variable.

6.0 VECTOR AND SCALAR

This topic explains the basic operations of vector and scalar quantities including
their use in solving problems. This topic also explains the method for determining
angle between two vectors as well as the characteristics of triple vector and scalar
products.