This belongs to Scalars and Vectors

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2. This is a REGULAR CLASS. Please make a
note by writing down what you have
seen here.
Make a separate note for CURRENT
ELECTRICITY.

3. UDAY KHANAL
Department of Physics
CCRC
3

4. Resolution of vectors
It is the process of splitting of a vector into two or more vectors. The vectors thus obtained are
called components.

5. Rectangular resolutions
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Q
R
O
𝜃
𝐴
Value of Ax and Ay
𝑖𝑛 ∆ OPQ
𝑐𝑜𝑠𝜃 =
𝑂𝑄
𝑂𝑃
=
𝐴𝑥
𝐴
→ 𝐴𝑥 = 𝐴𝑐𝑜𝑠𝜃
In rectangular
resolution two
components
must be
perpendicular.
The component on which angle is
formed is termed as cos component
𝑎𝑛𝑑,
𝑠𝑖𝑛𝜃 =
𝑃𝑄
𝑂𝑃
=
𝐴𝑦
𝐴
→ 𝐴𝑦 = 𝐴𝑠𝑖𝑛𝜃
From the side of the triangle law;
𝑂𝑃 = 𝑂𝑄+ 𝑄𝑃
𝑜𝑟, 𝐴 = 𝐴𝑥 + 𝐴𝑦
= 𝑖𝐴𝑥 + 𝐽𝐴𝑦
Thus given vector = sum of components
From the side of magnitude;
A = 𝐴𝑥
2 + 𝐴𝑦2 + 2𝐴𝑥𝐴𝑦𝑐𝑜𝑠90°
= 𝐴𝑥
2 + 𝐴𝑦2
∴ 𝑀𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒 𝑐𝑎𝑛 𝑏𝑒 𝑜𝑏𝑡𝑎𝑖𝑛𝑒𝑑 𝑓𝑟𝑜𝑚
𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑜𝑢𝑠 𝑡ℎ𝑒𝑜𝑟𝑒𝑚
From the side of Direction;
𝑡𝑎𝑛𝜃 =
𝐴𝑦
𝐴𝑥
In the component notation,
𝑨 = { 𝐴𝑥
2 + 𝐴𝑦2 ,
𝐴𝑦
𝐴𝑥
}
Magnitude Direction
Vector
𝐴𝑥 = 𝑖𝐴𝑥
𝐴
𝑦
=
𝐽𝐴
𝑦
x
y
X-Component
Y-Component

6. In 3 Dimension:
𝐴 = 𝐴𝑥 + 𝐴𝑦+ 𝐴𝑧
= 𝑖𝐴𝑥 + 𝐽𝐴𝑦+ 𝑘𝐴𝑧
𝐴 = { 𝐴𝑥
2 + 𝐴𝑦
2 + 𝐴𝑧2 , 𝑡𝑎𝑛𝜃 =
𝐴𝑦
𝐴𝑥
}
Magnitude Direction

7. Then
𝑎 + 𝑏 + 𝑐 = 𝑎𝑥 + 𝑏𝑥 + 𝑐𝑥 𝑖 + 𝑎𝑦 + 𝑏𝑦 + 𝑐𝑦 𝑗 + 𝑎𝑧 + 𝑏𝑧 + 𝑐𝑧 𝑘
And
If,
𝑎 = 𝑎𝑥 𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘 ,
𝑏 = 𝑏𝑥 𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘
𝑐 = 𝑐𝑥 𝑖 + 𝑐𝑦𝑗 + 𝑐𝑧𝑘
Note

8. 𝜃 𝜃
𝜃 𝜃
𝜃
𝜃
𝜃
𝜃
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A A
A A
A
A
A
A
𝐴𝑠𝑖𝑛𝜃
−𝐴𝑐𝑜𝑠𝜃
𝐴𝑠𝑖𝑛𝜃
𝐴𝑐𝑜𝑠𝜃
𝐴𝑐𝑜𝑠𝜃
𝐴𝑠𝑖𝑛𝜃
𝐴𝑠𝑖𝑛𝜃
𝐴𝑐𝑜𝑠𝜃
−𝐴𝑠𝑖𝑛𝜃
−𝐴𝑠𝑖𝑛𝜃 −𝐴𝑐𝑜𝑠𝜃
−𝐴𝑠𝑖𝑛𝜃
−𝐴𝑐𝑜𝑠𝜃 𝐴𝑐𝑜𝑠𝜃
−𝐴𝑠𝑖𝑛𝜃 −𝐴𝑐𝑜𝑠𝜃
30°
−𝑨𝒄𝒐𝒔𝟑𝟎°
𝒐𝒓, −𝑨𝒔𝒊𝒏𝟔𝟎°
𝒐𝒓, 𝑨𝒄𝒐𝒔𝟏𝟓𝟎°
A
x x x x
x
x
x
x
y y y y
y
y
y
y
y
x
Some Practices on Rectangular Components

11. 𝒐𝒓, −𝑶𝑪 𝒄𝒐𝒔𝟒𝟓°

13. 𝜽 = 𝐜𝐨𝐬
_𝟏(
𝟐
𝟑
)

16. Alternatively
𝛼 = tan
_1
𝐵𝑠𝑖𝑛90°
𝐴 + 𝐵𝑐𝑜𝑠90°

18. Three Forces of magnitude 2N, 3N, and 4N act in a plane at a point. The angle
between any two of their direction being 120°. Find the magnitude and direction
of their resultant.
2N
3N
4N
60°
x
y
60°
𝑅
𝜃
Work Out
Hints
𝑅𝑥 =. . . . . .
𝑅𝑦 =. . . . . . . . . .
𝑅 = 𝑅𝑥
2 + 𝑅𝑦2 . . . . . . . . .
𝜃 = tan
_1 𝑅𝑦
𝑅𝑥
.. . . . . . . . .

19. 10N
5N
15N
30°
x
y
45°
60°
Work Out
Hints
𝑅𝑥 =. . . . . .
𝑅𝑦 =. . . . . . . . . .
𝑅 = 𝑅𝑥
2 + 𝑅𝑦2 . . . . . . . . .
𝜃 = tan
_1 𝑅𝑦
𝑅𝑥
.. . . . . . . . .
y
2 m
3 m
5 m
27°
x

20. The essence of SCIENCE: ask an impertinent question, and
you are on the way to a pertinent answer.

21. DID YOU ENJOYE THE CLASS?
Leave your valuable suggestions so that I will be
better for you all in the next class. Your
suggestions are highly appreciated.
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