VECTOR
OPERATION
ANALYTICAL
GRAPHICAL
METHOD
GROUP 2:
BA L DU EZA , E RW I N
CU DA L , A N G E L E M E L INE
L I N DE, DI E T HER
M E JI A , DE M E TRIO
IN
AND
VECTORS
- A vector is an object that has both
magnitude and direction. This can be
represented by a line segment where
the length is the magnitude and the
arrow indicates the direction.
VECTORS
- In vector, we use the terms:
 Resultant vector – the sum of all vectors
 Scaling – line transformation that enlarges or diminishes a
vector
VECTOR OPERATION IN ANALYTICAL
METHOD
- Analytical method of vector addition and subtraction employs geometry and
simple trigonometry. This method is more concise, accurate and precise but,
are limited only by the accuracy and precision with which physical quantities
are known.
- In analytical method, we use the:
a. Pythagorean Theorem
b. Trigonometric Function
c. Component Method
1. A man runs 50m east
then 50m north.
Determine the resultant
vector.
a. By Pythagorean Theorem:
𝑐 2 = 𝑎2 + 𝑏 2
𝑐 2 = 502 + (502 )
𝑐2 =
C = 70.71 m
502 + (502 )
Vector Addition in Analytical Method
1. A man runs 50m east
then 50m north.
Determine the resultant
vector.
b. Solving Ɵ, using Trigonometric
function.
𝑎
−1
Ɵ = tan ( )
𝑏
50𝑚
−1
Ɵ = tan (
)
50𝑚
Ɵ = 𝟒𝟓𝒐
𝑹 = 𝟕𝟎. 𝟕𝟏𝒎, 𝟒𝟓𝟎 𝑵 𝒐𝒇 𝑬
(Ans)
2. Two person are pulling a box. The
first person applies 6N at NE
direction and the second applies 5N
at 600 N of W.
Scaling: 1N = 1cm
6N = 6cm
5N = 5cm
• We can’t solve this problem directly
using Trigonometric function and
Pythagorean theorem. However, we
can use the Component Method
where we rewrite the vectors into its
components.
a. Horizontal component:

𝑉𝑥 = 𝑉 cos Ɵ
b. Vertical component:

𝑉𝑦 = 𝑉 sin Ɵ
VECTOR
Ɵ
VX
VY
6N
450
+ 4.24
+ 4.24
5N
600
-2.5
+ 4.33
𝑉𝑥 = 1.74 𝑁
𝑉𝑦 = 8.57𝑁
• To get the resultant, we can now
use the Pythagorean theorem:
𝐶=
𝑉𝑥 +
𝑉𝑦
𝐶 = 1.74𝑁 2 + 8.57𝑁 2
C = 8.74N
• And use the Trigonometric
Function to get the Ɵ:
Ɵ=
tan−1 (
𝑉𝑥
)
𝑉𝑦
8.57𝑁
Ɵ=
)
1.74𝑁
Ɵ = 𝟕𝟖𝟎
tan−1 (
R = 8.74N, 780 N of E
(Ans.)
1. A man runs 40m
west then 40m south.
Determine the resultant
vector.
a. By Pythagorean Theorem:
𝑐 2 = 𝑎2 + 𝑏 2
𝑐 2 = −402 + (−402 )
𝑐2 =
C = 56.57 m
−402 + (−402 )
Vector Subtraction in Analytical Method
1. A man runs 40m
west then 40m south.
Determine the resultant
vector.
b. Solving Ɵ, using Trigonometric
function.
𝑎
−1
Ɵ = tan ( )
𝑏
−40𝑚
−1
Ɵ = tan (
)
−40𝑚
Ɵ = 𝟒𝟓𝒐
𝑹 = 𝟓𝟔. 𝟓𝟕𝒎 𝒂𝒕 𝟒𝟓𝟎 𝑺𝑾
(Ans)
VECTOR OPERATION IN GRAPHICAL METHOD
- Graphical method of vector in addition and subtraction made use of tools such
as rulers and protractors where parts of the vector is retained, because it is
represented by arrows for visualization. However, this method is limited by the
accuracy with which the drawing or graphs can be made.
- In this method, we can use the:
a. Head-to-tail or Triangle method – use for 2 given vectors.
b. Polygon Method – used for more than 2 given vectors.
1. A man runs 50m east then
50m north. Determine the
resultant vector.
Scaling: 10m = 1 cm
50 m x 1cm/10m = 5cm
 Using
a
ruler,
magnitude is 7.1cm
Vector Addition in Graphical Method
the
10𝑚
7.1 𝑐𝑚 𝑥
= 𝟕𝟏𝒎
1𝑐𝑚
5cm
 Using the protractor, the Ɵ
is, 450
 Thus, the R = 71m at 450
NE.
5cm
2. A bird flies 80m due East, 100m due
NE, 110 m at 300 N of W and 160m
at 200 S of W. Where is the bird?
Scaling: 20m = 1cm
80m = 4cm
100m = 5cm
110m = 5.5cm
160m = 8cm
 Using the ruler, we calculated
the magnitude as 5.9cm
20𝑚
5.9𝑐𝑚 𝑥
= 𝟏𝟏𝟖𝒎
1𝑐𝑚
 Using the protractor, the Ɵ is
36.50
 Thus, the bird can be found
118m at 36.50 N of W.
1. A man runs 40m west then
40m south. Determine the
resultant vector.
Scaling: 10m = 1 cm
1𝑐𝑚
40𝑚 𝑥 ( ) = 4cm
10𝑚
 Using
a
ruler,
magnitude is 5.7cm
the
10𝑚
5.7 𝑐𝑚 𝑥
= 𝟓𝟕𝒎
1𝑐𝑚
 Using the protractor, the Ɵ
is, 450
 Thus, the R = 57m at 450
SW.
Vector Subtraction in Graphical Method