Vectors

1. EMA 310: Vectors and Mechanics
Unit 2: Algebra of vectors
Dr. Isaac Benning
Dept. of Maths & ICT Education
Faculty of Science and Technology Education
University of Cape Coast

2. Learning objectives
By the end of this unit, you should be able to:
1. find the resultant of given vectors.
2. add vectors using parallelogram and triangle laws of addition.
3. establish and use properties of addition of vectors.

3. Addition of Vectors
a. The triangle law of vector addition
The sum of vectors, 𝐴𝐵 and 𝐵𝐶 is defined as the single or equivalent or resultant vector
𝐴𝐶. We write this as:
𝑨𝑩 + 𝑩𝑪 = 𝑨𝑪
or
𝒂 + 𝒃 = 𝒄
This rule is known as the triangle law of vector addition.

4. Addition of Vectors
Thus, to find the sum of two vectors , 𝒂 and 𝒃, we draw a chain, starting the second where the
first ends: c is given by the single vector joining the start of the first to the end of the second.

5. Addition of Vectors
a. The triangle law of vector addition

6. Addition of Vectors
b. The parallelogram law of vector addition
If two vectors a and b have a common initial point and can be represented in a magnitude
and direction by two sides of 𝑂𝐴 and 𝑂𝐵 parallelogram 𝑂𝐴𝑃𝐵, then their sum, 𝑎 + 𝑏 is
represented in magnitude and direction by the diagonal 𝑂𝑃.
Click to see animation

7. Properties of Addition Vectors

8. Activity (Individual)
Find the vector sum.
1. 𝐴𝐵 + 𝐵𝐶 + 𝐶𝐷 + 𝐷𝐸 + 𝐸𝐹 + 𝐹𝐺 =
2. 𝐴𝐾 + 𝐾𝐿 + 𝐿𝑃 + 𝑃𝑄 =
3. 𝐴𝐵 − 𝐶𝐵 + 𝐶𝐷 − 𝐸𝐷=
4. 𝐴𝐶 + 𝐶𝐿 − 𝑀𝐿 =

9. Multiplication of a vector by a scalar
If k a real number (scalar) other than zero
and 𝒖. Then 𝑘𝒖 is a vector defined as
follows:
1. If k > 0, 𝑘𝒖 is a vector in the same
direction as 𝒖 and 𝑘 times as long.
2. If k < 0, 𝑘𝒖 is a vector in the opposite
direction to 𝒖 and (−𝑘) times a long.
3. If two vectors are parallel, then one is a
scalar multiple of the other.
𝒖
𝟐𝒖
−3
2
𝒖

10. Activity (Whole class discussion)
If the vector 𝑐 = 𝑎 + 2𝑏 and 2𝑐 = 𝑎 – 3𝑏,
show that:
(i) vectors a and c have the same direction
(ii) vectors a and b have opposite direction.

11. Activity (Whole class discussion)
(i) 𝑎 + 2𝑏 = 𝑐 … … … … … … … . ((1)
a − 3b = 2c … … … … … … … (2)
3a + 6b = 3c … … … … … … . (3)
2𝑎 − 6𝑏 = 4𝑐 … … … … … … . (4)
5𝑎 = 7𝑐
𝑎 =
7
5
𝑐
Thus, vector a is a scalar multiple of vector c. since the
scalar is greater than zero, vectors a and c have the
same direction.

12. Activity (Whole class discussion)
(ii) (1) × 2; 2𝑎 + 4𝑏 = 2𝑐 … … … … … … … … . . (5)
𝑎 − 3𝑏 = 2𝑐 … … … … … … . . (2)
∴ 𝑎 + 7𝑏 = 0
𝑎 = −7𝑏
Thus, vector a and b have opposite directions.

13. Activity (Whole class discussion)
ABCD is a quadrilateral, with G and H as the midpoints of
DA and BC respectively. Show that 𝐴𝐵 + 𝐷𝐶 = 2𝐺𝐻

14. Activity (Whole class discussion)
ABCD is a quadrilateral, with G and H as the midpoints of DA and BC respectively.
Show that 𝐴𝐵 + 𝐷𝐶 = 2𝐺𝐻
𝐴𝐵 = 𝐴𝐺 + 𝐺𝐻 + 𝐻𝐵
𝐷𝐶 = 𝐷𝐺 + 𝐺𝐻 + 𝐻𝐶 𝑠𝑖𝑚𝑖𝑙𝑎𝑟𝑙𝑦
𝐴𝐵 + 𝐷𝐶 = 𝐴𝐺 + 𝐺𝐻 + 𝐻𝐵 + 𝐷𝐺 + 𝐺𝐻 + 𝐻𝐶
= 2 𝐺𝐻 + 𝐴𝐺 + 𝐷𝐺 + 𝐻𝐵 + 𝐻𝐶
But 𝐷𝐺 = −𝐴𝐺 (𝐷 𝑖𝑠 𝑎 𝑚𝑖𝑑𝑝𝑜𝑖𝑛𝑡 𝑜𝑓 𝐴𝐷)
𝐻𝐶 = −𝐻𝐵
𝐴𝐵 + 𝐷𝐶 = 2𝐺𝐻 + 𝐴𝐺 − 𝐴𝐺 + 𝐻𝐵 − 𝐻𝐵 = 2𝐺𝐻

15. Exercise
1. Show that if 𝑚 =
1
2
𝑎 + 𝑏 then m is the midpoint of AB.
2. 𝑂𝐴𝐵 is triangle, where O is the origin, let a and b be the positive vectors of
A and B respectively. If 𝑋 is a point on 𝐴𝐵 such that 𝐴𝑋 = 2𝑋𝐵 and Y is the
midpoint of OX, show that 𝐵𝑌 =
1
6
𝑎 −
2
3
𝑏.
3. Points 𝐿, 𝑀, 𝑁 are midpoints of the sides 𝐴𝐵, 𝐵𝐶, 𝐶𝐴 of a triangle 𝐴𝐵𝐶.
Show that,
a. 𝐴𝐵 + 𝐵𝐶 + 𝐶𝐴 = 0
b. 2𝐴𝐵 + 3𝐵𝐶 + 𝐶𝐴 = 2𝐿𝐶
c. 𝐴𝑀 + 𝐵𝑁 + 𝐶𝐿 = 0