Simple Math

1. VECTOR
ALGEBRA
[CLO2]
Norlianah Mohd Shah

2. LEARNING
OUTCOMES
 At the end of this topic, students should be
able to:
 Vectors in the plane
 Dot product and its use defining physical quantities
 Cross product and its use in defining angular velocity,
motion of charged particles in electromagnetic field
and their geometrical applications

3. Introduction
The fundamentals of vectors
Operations on vectors
Scalar (dot) products of vectors
Vector (cross) products of vectors

6. 6
A scalar is a quantity
with magnitude but
without direction.
Example: A speed of
10km/h.
What is Scalar?

7. 7
A vector is a quantity
that has both magnitude
and direction.
Example: A velocity of
20km/h at 20 degree
What is Vector?

8. 8
01Diagrams and notations.
02Illustration of vector in 2D and 3D
graph.
03Types of vector.
04Magnitude of vector.

9. 9
06Operations on vectors.
07Resultant of vectors.
08Unit vector.

10. 10
09Scalar (Dot) products of vectors.
10Vector (cross) products of vectors.

11. Magnitude
Head
Tail Direction
A bold capital letter
for the name of the
vector
An arrow above the
vector name
Direction of the point Matrix form from the
origin (0, 0) in the
direction of the point
(a, b)
𝑨 𝑨 (𝑎, 𝑏) 𝑎
𝑏
Vector notation
Exampl
e
𝑂𝐴 𝐴𝐵 𝒂 𝑎
4
5
(4, 5) 4𝑖 + 5𝑗 4𝑖 + 5𝑗
−1
2
−5
(−1, 2, −5) −𝑖 + 2𝑗 − 5𝑘 −𝑖 + 2𝑗 − 5𝑘

12. 2D vector 3D vector

13. Type of vector Example
Equal vector 𝐹1 = 𝐹2
Negative vector
Position vector
Free vector
Parallel vector
Resultant vector 𝐹𝑅 = 𝐹1 + 𝐹2
Equilibrium vector 𝐹𝐸 = −𝐹𝑅
Co-linear vector
O
A
𝒂
A B

14. Magnitude:
The length of a vector
2D vector, 𝒗 = (𝑥, 𝑦)
𝒗 = 𝑥2 + 𝑦2
3D vector, 𝒗 = (𝑥, 𝑦, 𝑧)
𝒗 = 𝑥2 + 𝑦2 + 𝑧2

15. EXAMPLE
1
Find the magnitude of the following vectors:
1. 𝑨 = −5,2
2. 𝑩 = (5, 6, 3)

16. VECTOR LAWS FOR ADDITION &
SUBTRACTION
 Commutative Law 𝒂 + 𝒃 = 𝒃 + 𝒂
 Associative Law
𝒂 + 𝒃 + 𝒄 = 𝒂 + (𝒃 + 𝒄)
 Distributive Law
𝜆 𝒂 + 𝒃 = 𝜆𝒂 + 𝜆𝒃
 Subtraction
𝒂 − 𝒃 = 𝒂 + (−𝒃)

17. EXAMPLE 2
 Given that vector 𝒗 = −3𝑖 + 2𝑗 and 𝒘 = 5𝑖 − 9𝑗. Express the
following in terms of i and j.
a) 𝒗 + 𝒘
b) 𝒗 − 𝒘

18. Example 3
Given that vector 𝒖 = (−5,4) and 𝒘 =
(1, −1). Find:
a) 3𝒖
b) −7𝒘

19. Two vectors A and B:
Vector 𝐴 = 𝑂𝐴
Vector 𝐵 = 𝑂𝐵
Resultant 𝐴𝐵 = 𝐴𝑂 + 𝑂𝐵 = −𝑂𝐴 + 𝑂𝐵

20. Resultant of vectors
 Let 𝑂𝐴 = 𝑎 and 𝑂𝐵 = 𝑏.
O
A
B

21. Given that the vectors 𝐴 = 2𝑖 + 3𝑗 − 𝑘, 𝐵 = 4𝑖 − 3𝑗 + 2𝑘 and 𝐶 = 𝑖 + 2𝑗 − 3𝑘. Find
each of the following:
 𝐴𝐵
 𝐵𝐶

22.  Theorem (Unit Vector 𝒗)
 If 𝒗 is a non-null vector and if 𝒗 is the unit vector having the same direction as v, then
 𝒗 =
𝒗
𝒗
 𝑈𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 =
𝑣𝑒𝑐𝑡𝑜𝑟
𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑣𝑒𝑐𝑡𝑜𝑟
 𝒗 =
𝑣
𝑣
=
𝑥𝑖+𝑦𝑗+𝑧𝑘
𝑥2+𝑦2+𝑧2
=
𝑥
𝑥2+𝑦2+𝑧2
𝑖 +
𝑦
𝑥2+𝑦2+𝑧2
𝑗 +
𝑧
𝑥2+𝑦2+𝑧2
𝑘
𝒗

23.  Given that vector 𝑦 = 3𝑖 − 𝑗 + 7𝑘. Find the unit vector of 𝑦.

24.  Given that vector 𝐾 = 2𝑖 + 3𝑗 − 𝑘 and 𝐿 = 4𝑖 − 3𝑗 + 2𝑘. Find the unit vector of 𝐾𝐿.

25.  The dot product of two vectors a= 𝑎1, 𝑎2, 𝑎3 and b= 𝑏1, 𝑏2, 𝑏3 is defined by
 𝐚 ∙ 𝐛 = 𝑎1, 𝑎2, 𝑎3 ∙ 𝑏1, 𝑏2, 𝑏3
= 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3
 Note that:
The dot product of two vectors is a scalar (number, not a vector)

26.  Given that a vector 𝒂 = 5𝑖 + 3𝑗 − 2𝑘 and 𝒃 = 8𝑖 − 9𝑗 + 11𝑘. Find 𝒂 ∙ 𝒃.

28.  Two vectors A and B measure 8 units and 5 units length respectively. If the angle
at which they are inclined with each other is 60°, determine the dot product.

29. Find the angle between the vectors 𝒂 = 2𝑖 + 3𝑗 + 5𝑘 and 𝒃 = 𝑖 − 2𝑗 + 3𝑘.

33.  For two vectors a= 𝑎1, 𝑎2, 𝑎3 and b= 𝑏1, 𝑏2, 𝑏3 we define the cross product of a
and b to be
𝐚 × 𝐛 =
𝐢 𝐣 𝐤
𝑎1 𝑎2 𝑎3
𝑏1 𝑏2 𝑏3
=
𝑎2 𝑎3
𝑏2 𝑏3
𝐢 −
𝑎1 𝑎3
𝑏1 𝑏3
𝐣 +
𝑎1 𝑎2
𝑏1 𝑏2
𝐤
 Note that:
The cross product of two vectors is another vector.

34.  𝐚 × 𝐛 = −𝐛 × 𝐚 anti-commutativity
 𝑑𝐚 × 𝐛 = 𝑑 𝐚 × 𝐛 = 𝐚 × (𝑑𝐛)
 𝐚 × 𝐛 + 𝐜 = 𝐚 × 𝐛 + 𝐚 × 𝐜 distributive law
 𝐚 ∙ (𝐛 × 𝐜) = (𝐚 × 𝐛) ∙ 𝐜 scalar triple product
 𝐚 × 𝐛 × 𝐜 = 𝐚 ∙ 𝐜 𝐛 − 𝐚 ∙ 𝐛 𝐜 vector triple product
 if two vectors a and b are parallel, then 𝐚 × 𝐛 = 0
Theorem 3.1 Let 𝜃 be the angle between nonzero vectors a and b. Then,
𝐚 × 𝐛 = 𝐚 𝐛 sin 𝜃

35. Any nonzero vectors a and b, as long as a and b are
not parallel, they form two adjacent sides of
parallelogram. Notice that the area of parallelogram is
given by the product of the base and altitude.
Area= base altitude
 = 𝐛 𝐚 sin θ = |𝐚 × 𝐛|

36.  Calculate the cross product between vectors 𝒂 = (3, −3,1) and 𝒃 = (4, 9, 2).

37.  Given that vectors 𝑨 = 2𝑖 + 3𝑗 + 4𝑘, 𝑩 = 𝑖 − 2𝑗 + 3𝑘 and 𝑨 × 𝑩 = 17𝑖 − 2𝑗 − 7𝑘.
 Find the angle between vectors A and B.

38.  Calculate the area of parallelogram with vectors p = (-1, -5, -12) and q = (3, 5, 1) as
its sides.

39. 1. Given that vectors 𝑎 = (2, −1) and 𝑏 = (3, 4). Calculate 𝑎 ∙ 𝑏. Hence, find the angle between 𝑎
and 𝑏.
2. Given that vectors 𝑎 = 3𝑖 + 𝑥𝑗 − 2𝑘 and 𝑏 = 1 − 𝑥 𝑖 − 3𝑗 + 4𝑘, find 𝑥 if 𝑎 is perpendicular to 𝑏.
3. Given that points 𝑋 = 2𝑖 + 5𝑗 − 4𝑘, 𝑌 = 3𝑖 + 4𝑗 + 3𝑘, and 𝑍 = −4𝑖 + 5𝑗 + 𝑘. Find 𝑋𝑌 ∙ 𝑋𝑍.
4. Given vectors 𝐴 = 2𝑖 + 3𝑗 + 6𝑘 and 𝐵 = 𝑖 − 𝑗 + 2𝑘. Find the area of the parallelogram spanned
by the vectors.
5. Given that vectors 𝑎 = 𝑖 + 2𝑗 + 3𝑘 and 𝑏 = −𝑖 + 3𝑗 − 𝑘:
a) Find 𝑎 × 𝑏.
b) Prove that 𝑎 × 𝑏 is a vector which is perpendicular to a vector 𝑎.
c) Given that vector 𝑐 = 2𝑖 + 4𝑗 + 6𝑘, prove that 𝑎 and 𝑐 are parallel.

40. QUIZ 19 APR 2021
1. Given vector 𝒖 = 9,8,7 , 𝒗 = (3,4,5) and 𝒘 = (8,5,3). Calculate
2𝑢 + 2w.
2. Find the dot product and an angle between vectors 𝑢 = 2,2,3
and 𝑣 = 1,3,6
3. Find the dot product and an angle between vectors 𝑢 =
−1, −2,3 and 𝑣 = (5,6,1)
4. Find the dot product and an angle between vectors 𝑢 =
−1, −2, −3 and 𝑣 = (4,4, −4)
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41. TASK 3
1. Find the cross product and an angle between vectors 𝑢 = 1,2,4
and 𝑣 = (5,6,3)
2. Find the cross product and an angle between vectors 𝑢 =
−1, −2, −3 and 𝑣 = (4,4, −4)
 Answer: 1. 𝟒𝟎. 𝟖𝟓° 2. 𝟗𝟎°
42

42. THANK
YOU