This belongs to Scalars and Vectors

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1

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

5. Lame's Theorem
If three vectors acts simultaneously on a particle and system is in the
equilibrium, then
𝛼
𝛽
𝛾
𝐴
𝐵
𝐶 𝐴
𝑠𝑖𝑛𝛼
=
𝐵
𝑠𝑖𝑛𝛽
=
𝐶
sin𝛾

6. Multiplications Between Two Vectors
. ; dot; Scalar product → 𝑆𝑐𝑎𝑙𝑎𝑟 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
; cross; Vector product → 𝑉𝑒𝑐𝑡𝑜𝑟 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦
X

7. Scalar Product (or, dot Product)
Where,
A – magnitude of 𝐴
B- Magnitude of 𝐵
And 𝜃 𝑖𝑠 𝑡ℎ𝑒 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐴 𝑎𝑛𝑑 𝐵
𝜃
.
.
.
.
.
.
𝐴
𝐵
Acos𝜃
𝐴. 𝐵 = 𝐴𝐵𝑐𝑜𝑠𝜃
= magnitude of one vector x Scalar component of the
second vector along the direction of first vector
Scalar product between 𝐴 𝑎𝑛𝑑 𝐵 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠
𝐴. 𝐵 = 𝐴𝐵𝑐𝑜𝑠𝜃 ;

8. Some points on Dot Product
𝑨. 𝑩 = 𝑨𝑩𝒄𝒐𝒔𝜽
𝐴. 𝐵 = 𝐵. 𝐴 ∶ 𝐶𝑜𝑚𝑚𝑢𝑡𝑎𝑡𝑖𝑣𝑒
𝐴. 𝐵 + 𝐶 = 𝐴. 𝐵 + 𝐴. 𝐶 Distributive
𝐼𝑓 𝜃 = 0;
𝐴. 𝐵 = 𝐴𝐵𝑐𝑜𝑠0 = 𝐴𝐵
𝐼𝑓 𝜃 = 90°;
𝐴. 𝐵 = 𝐴𝐵𝑐𝑜𝑠90° = 0
𝐼𝑓 𝜃 = 180°;
𝐴. 𝐵 = 𝐴𝐵𝑐𝑜𝑠180° = −𝐴𝐵
𝐴. 𝐴 = 𝐴. 𝐴 𝑐𝑜𝑠0° = 𝐴2
𝐴. 𝐴 = 1.1𝑐𝑜𝑠0° = 1
𝑖. 𝑖 = 1
𝑗. 𝑗 = 1
𝑘. 𝑘 = 1
𝑖. 𝑗 = 0 = j. 𝑖
𝑗. 𝑘 = 0 = 𝑘. 𝑗
𝑘. 𝑖 = 0 = 𝑖. 𝑘

9. 𝑎 = 𝑎𝑥 𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘 ,
𝑏 = 𝑏𝑥 𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘
If,
Example:
If,
𝑎 = 3𝑖 + 4𝑗 and
𝑏 = −2𝑖 + 3𝑘, 𝑓𝑖𝑛𝑑 ′𝜃′ [𝑎𝑛𝑠 ; 109°]
𝑎. 𝑏 = 𝑎𝑥 𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘 . (𝑏𝑥 𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘)
= 𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧
𝑎 = 𝑎 = 𝑎𝑥
2 + 𝑎𝑦2 + 𝑎𝑧2
𝑏 = 𝑏 = 𝑏𝑥
2 + 𝑏𝑦2 + 𝑏𝑧2
𝑛𝑜𝑤, 𝑎. 𝑏 = 𝑎𝑏𝑐𝑜𝑠𝜃
𝑐𝑜𝑠𝜃 =
𝑎. 𝑏
𝑎𝑏
=
𝑎𝑥𝑏𝑥 + 𝑎𝑦𝑏𝑦 + 𝑎𝑧𝑏𝑧
𝑎𝑥
2 + 𝑎𝑦2 + 𝑎𝑧2 𝑏𝑥
2 + 𝑏𝑦2 + 𝑏𝑧2

10. Problem

11. Vector Product (or, Cross Product)
= X× 𝒀
Right hand co-ordinate system

14. Defination:
𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝐴 𝑎𝑛𝑑 𝐵 𝑖𝑠 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑎𝑠;
𝑨 × 𝑩 = 𝑨𝑩𝒔𝒊𝒏𝜽 𝒏 ,
Where, 𝑛 − 𝑖𝑠 𝑎 𝑢𝑛𝑖𝑡 𝑣𝑒𝑐𝑡𝑜𝑟 𝑝𝑒𝑟𝑝𝑒𝑛𝑑𝑖𝑐𝑢𝑙𝑎𝑟 𝑡𝑜 𝑡ℎ𝑒 𝑝𝑙𝑎𝑛𝑒 𝑐𝑜𝑛𝑡𝑎𝑖𝑛𝑖𝑛𝑔 𝑏𝑜𝑡ℎ 𝐴 𝑎𝑛𝑑 𝐵.

15. O
M Q
P
H𝑒𝑟𝑒,
𝐴 × 𝐵 = 𝐴𝐵𝑠𝑖𝑛𝜃
= 𝑂𝑃 𝑂𝑀 𝑠𝑖𝑛𝜃
= 𝑂𝑃 𝑀𝑁 ∴
𝑀𝑁
𝑂𝑀
= 𝑠𝑖𝑛𝜃
= 𝑏𝑎𝑠𝑒 × ℎ𝑒𝑖𝑔ℎ𝑡 𝑜𝑓 𝑡ℎ𝑒 𝑂𝑃𝑄𝑀
= 𝑎𝑟𝑒𝑎 𝑜𝑓 𝑡ℎ𝑒 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑔𝑟𝑎𝑚
= 𝑪 = 𝑨𝑩𝒔𝒊𝒏𝜽 𝒏
Physical Significance of Cross Product
N

16. Some points on Cross Product
• 𝒂 × 𝒂 = 𝒂𝒂𝒔𝒊𝒏𝟎° 𝒏 = 𝟎
• 𝒂 × 𝒃 + 𝒄 = 𝒂 × 𝒃 + 𝒂 × 𝒄 ∶ 𝒅𝒊𝒔𝒕𝒓𝒊𝒃𝒖𝒕𝒊𝒗𝒆
𝑖𝑓 𝜃 = 0°
𝑎 × 𝑏 = 𝑎𝑏𝑠𝑖𝑛0° 𝑛 = 0
𝑖𝑓 𝜃 = 90°
𝑎 × 𝑏 = 𝑎𝑏𝑠𝑖𝑛90° 𝑛 = 𝑎𝑏 𝑛
𝑖𝑓 𝜃 = 180°
𝑎 × 𝑏 = 𝑎𝑏𝑠𝑖𝑛180° 𝑛 = 0
• 𝑖 × 𝑖 = 0
𝑗 × 𝑗 = 0
𝑘 × 𝑘 = 0
• 𝒂 × 𝒃 = −𝒃 × 𝒂 ∶ 𝒏𝒐𝒏 𝒄𝒐𝒎𝒎𝒖𝒕𝒂𝒕𝒊𝒗𝒆
𝑖 × 𝑗 = 𝑘
𝑗 × 𝑘 = 𝑖
𝑘 × 𝑖 = 𝑗
𝑗 × 𝑖 = −𝑘
𝑘 × 𝑗 = −𝑖
𝑖 × 𝑘 = −𝑗
Orthogonal Triad of unit vector
𝒂 × 𝒃 = 𝒂𝒃𝒔𝒊𝒏𝜽 𝒏

17. 𝑎 × 𝑏 =
𝑖 𝑗 𝑘
𝑎𝑥 𝑎𝑦 𝑎𝑧
𝑏𝑥 𝑏𝑦 𝑏𝑧
= 𝑖
𝑎𝑦 𝑎𝑧
𝑏𝑦 𝑏𝑧
+ 𝑗
𝑎𝑧 𝑎𝑥
𝑏𝑧 𝑏𝑥
+ 𝑘
𝑎𝑥 𝑎𝑦
𝑏𝑥 𝑏𝑦
= 𝑖 𝑎𝑦𝑏𝑧 − 𝑏𝑦𝑎𝑧 + 𝑗 𝑎𝑧𝑏𝑥 − 𝑏𝑧𝑎𝑥 + 𝑘(𝑎𝑥𝑏𝑦 − 𝑏𝑥𝑎𝑦)
𝑎 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘
𝑏 = 𝑏𝑥𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘
If,
Then,
Note:
Alternatively

18. Problem

19. Problem

20. Problem

21. 𝑎 = 2𝑖 + 3𝑗 + 5𝑘
𝑏 = 3𝑖 + 4𝑗 + 6𝑘
If,
𝐹𝑖𝑛𝑑, 𝑎 × 𝑏, 𝑎 × 𝑏 , 𝑑𝑖𝑟𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 × 𝑏 𝑖. 𝑒 𝑛 , a, b, sin𝜃,
𝑎𝑛𝑑 𝑎𝑛𝑔𝑙𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑎 𝑎𝑛𝑑 𝑏
Work Out

22. 𝑎 = 𝑎𝑥𝑖 + 𝑎𝑦𝑗 + 𝑎𝑧𝑘
𝑎𝑛𝑑 𝑏 = 𝑏𝑥𝑖 + 𝑏𝑦𝑗 + 𝑏𝑧𝑘 are
parallel
Two vectors
If,
|𝑎 × 𝑏| = 0
→
𝑎𝑥
𝑏𝑥
=
𝑎𝑦
𝑏𝑦
=
𝑎𝑧
𝑏𝑧
Remember:

23. 𝐴. 𝐵 × 𝐶 → 𝑆𝑐𝑎𝑙𝑎𝑟𝑡𝑟𝑖𝑝𝑝𝑙𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡
≡ 𝑉𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑎 𝑝𝑎𝑟𝑎𝑙𝑙𝑒𝑙𝑜𝑝𝑖𝑝𝑒𝑑
𝐴. 𝐵 × 𝐶 = 𝐵. 𝐶 × 𝐴 = 𝐶. 𝐴 × 𝐵
𝐴. 𝐵 × 𝐶 = 𝐴𝐵𝐶 =
𝐴𝑥 𝐴𝑦 𝐴𝑧
𝐵𝑥 𝐵𝑦 𝐵𝑧
𝐶𝑥 𝐶𝑦 𝐶𝑧
For coplanar vector 𝐴𝐵𝐶 = 0
Remember:

24. Exercises

25. Mind wash

26. Mind wash

27. Mind wash

28. Mind wash

29. Mind wash

30. Mind wash

31. Mind wash

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

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