Unit 2.3 Components of Vector

2.3 COMPONENTS OF A VECTOR

After completing the unit, students should be
able to :
1. compute the addition of a vector in two and

three dimensions (C1).
2. find the magnitude and unit vectors of two

and three dimensions (C3).

2.3.1 REPRESENT VECTORS IN CARTESIAN PLANE

The components of vectors can be represented in rectangular coordinate systems. If
a vector v is positioned with its initial point at the origin, O in a rectangular coordinate
system, then its terminal point. P will have coordinates in the form of (a , b) or (a , b ,
c), depending whether the vector is in 2 dimensional or 3 dimensional space. The
coordinate systems is as similar as in Cartesians plane that has the vertical and
horizontal axis for the 2 dimensional vector.

2 dimensional space 3 dimensional space

z

y P P
v
b v c y
Oa x O b

a
x

For the vector components of v can be For the vector components of v can be

written in two forms written as

i) Coordinate form : v = (a, b) i) Coordinate form: v = (a, b)

ii) Unit Vector form : OP = ai+bj ii) Unit Vector form:

OP = ai+ bj + ck

**There is no restriction in writing the form of vector. It depends based on the
questions or personal preferences.

Example 2.3.1: To write vectors

Sketch and write in a form of vector given the vector have initial points at the origin
and state your vector, v in vector unit form.

1. A (6, 3) 3. C (-7, 8) y x
y O
i
O6
A
3
x

OA = 6i + 3j

2. B (5, -1, 3) 4. D (3, 1, -5)
Solutions Solutions
z
B
y
x O

OB =

2.3.2 DIRECTION BETWEEN TWO VECTORS,

2 Dimensional Space 3 Dimensional Space

yz

v v
O
w x y
O xw

Theorem:

For v = (a , b) and w = (p , q) For v = (a , b , c) and w = (p , q , r)

→ = (p − a,q − b) → = (p − a,q − b,r − c)

vw vw

Example 2.3.2 :

1. y 2.
B (-5 , 6) y
C (5 , 2)
A (4 , 5) O x

x D (3 , -5)

O

Calculate Calculate

→ →

a) AB a) CD

→ →

b) BA b) DC

Solution Solution

a) → = (− 5 − 4,6 − 5)

AB

= (− 9,1)

**remember to subtract the right vector
to the left one.

→

b) BA =

=

3. u = −3i − j + k and v = 4i + 8j + 2k . 4. m = 10i + j − 7k and p = 6i + 2j + 3k
Calculate Calculate

→ →

a) uv a) mp

→ →

b) vu b) pm

Solutions Solutions

a) → = 4 − (− 3)i + 8 − (− 3) j + (2 − 1)k

uv

= 7i + 11j + k

→

b) vu =

=

2.3.3 MAGNITUDE OF A VECTOR, |v|

A) Magnitude for one single vector, v

The idea for magnitude is related to right-angle triangle and the application of
Pythagoras Theorem. The calculation of hypotenuse is integrated in the calculation of
magnitude of vector.

The distance between initial and terminal points of a vector, v is called the length, the
norm or the magnitude**, and denoted by |v|. the calculation of magnitude is given
by :

**Throughout the unit, we will use the term magnitude.

2 Dimensional Space 3 Dimensional Space

y (a, b) z (a, b, c)
|v|
b |v| c
O x y
a O
a b

x

|v| = √ 2 + 2 |v| = √ 2 + 2 + 2

**Note that the calculation is derived from Pythagoras Theorem.

Example 2.3.3 : Calculate magnitude of a vector

1. Calculate the magnitude of v = (8, -6) 2. Calculate the magnitude of
v = (-4 , -3)
Solutions:
Solutions:
|v| = √ 2 + 2
= √82 + (−6)2 = √100 = 10

3. Calculate the magnitude of 4. Calculate the magnitude of
v = (12, -6, 4) v = (12, 14, -12)

Solutions: Solutions:

|v| = √ 2 + 2 + 2

B) Magnitude between two vectors, |v→w|
If v an w is two vectors, the magnitude between two vectors is given by

2 Dimensional Space 3 Dimensional Space

For v = (a , b) and w = (p , q) For v = (a , b , c) and w = (p , q , r)
|v→w| = √( − )2 + ( − )2 |v→w| = √( − )2 + ( − )2 + ( − )2

**Note that the calculation under the square root is from directions formula.

Example 2.3.4: Calculate magnitude between two vectors, |v→w|

1. Given vector v = (5, 6) and w = (9 , 9). 2. Vector m = (14, -18) and p = (21 , 6).
Calculate |m→p|
→

Calculate the magnitude of vw

Solutions: Solutions:
⃗v⃗⃗⃗u = (9 − 6,9 − 5) |m→p| =

= (3,4)

|⃗v⃗⃗⃗u| = √32 + 42

= √25

=5

**remember to subtract the right
vector to the left one.

1. v = (12, 3, 8) and w = (8, -5, 10). 2. m = -2i + 4j + 16k and p = 2i – 9k.
Calculate |p→m|
→
Solutions:
Calculate the magnitude of vw

Solutions:
⃗v⃗⃗w⃗⃗ = (8 − 12, −5 − 3,10 − 2)

= (4, −8,8)

|⃗v⃗⃗w⃗⃗ | = √42 + (−8)2 + 82

= √144
= 12

2.3.4 UNIT VECTOR,

A) Unit vector for a single vector.

The operation to find a unit vector, u from a non-zero vector, v is called normalizing
vector. The calculation of u is given by
ȗ v
= |v|


Example 2.3.5: Calculate unit vector, u from one single vector, v

1. Calculate the unit vector for 2. Calculate the unit vector for
v = (5, 12) v = (-24, 7)

Solutions: v Solutions: v
|v| |v|
ȗ = ȗ =

|⃗v| = √a2 + b2

= √52 + 122 = 13

ȗ = v
|v|
5 12
= 13 i + 13 j


**Note that the unit vector, u is written
in form of i, j and k.

3. Calculate the unit vector for 4. Calculate the unit vector for
v = (12, -6, 4) v = (12, 14, -12)

Solutions: v Solutions:
|v|
ȗ =

|⃗v| = √a2 + b2 + c2

B) Unit vector between two vectors.

The calculation of unit vector between two vectors is given by

̑ = →
|⃗ ⃗ ⃗⃗ ⃗ |

→
Example 2.3.6: Calculate unit vector, u between two vectors, vw

1. Vector v = (5, 6) and w = (9 , 9). 2. Vector m = (14, -18) and p = (21 , 6).

→ →

Calculate the unit vector of wv Calculate the unit vector of pm

Solutions: Solutions:
v→w = (6 − 9, 5 − 9)

= (−3, −4)
|v→w| = √(−3)2 + (−4)2

= √25

=5

û = 3 i − 4 j
5 5

**remember to subtract the right
vector to the left one.

3. v = (12, 3, 8) and w = (8, -5, 10). 4. m = -2i + 4j + 16k and p = 2i – 9k.

→ →

Calculate the unit vector of vw Calculate the unit vector of pm

Solutions: Solutions:
v→w = (8 − 12, −5 − 3,10 − 2)
= (4, −8,8)

|v→w| = √42 + (−8)2 + 82

= √144

= 12

û = 4 i − 8 j + 8 k
12 12 12

û = 1 i − 2 j + 2 k
3 3 3

Tutorial 2.3
1. Represent graphically the following vectors.

a. v = 5i – 4j
b. w = -2i – j
c. m = i – 2j – 7k
d. n = 6i + 6j + 6k

2. Calculate the direction of the following vectors.

a) A and B

i)

ii)

iii)
iv)

b) p and m
i)

ii)
iii)
iv)

3. Calculate the magnitude unit vectors for the given vector.

a. v and w

i) |v|

ii) |w|

iii) |v→w|

iv) Unit vectors in direction of v
v) Unit vectors in direction of w
vi) Unit vectors in direction of
vii) Unit vectors in direction of

3. Calculate the magnitude unit vectors for the given vector.

b. P and Q

i) | |

ii) | |

→

iii) | |
iv) Unit vectors in direction of

v) Unit vectors in direction of

vi) Unit vectors in direction of

vii) Unit vectors in direction of

y

v 12

-15 x
O
5
-9 w

4. a. State v and w coordinate form.

b. Calculate the

→

i. direction of vw

→

ii. -5I vw I

→

iii. unit vector in direction of vw

5. If m = 11i +13j − 2k and p = −3i + j +10k are two vectors in 3
dimensional space. Calculate the

→

a. direction of mp

→

b. magnitude of mp

→

c. unit vector in direction of mp