20240623100710281. Physics ll vector analysis

Vector

2. VECTORS

3. What
is Scalar?

4. (Cengthyf acar E (er)

ne +

5. of gold bar De

physical quantity magnitude

6. Time is 12.766)

^
physical quantity magnitude

7. Temperature 48686

physical quantity magnitude

8. A scalar is a physical quantity that
has only a magnitude.

Examples:
. Mass ・ Temperature
* Length ・ Volume
ㆍ Time « Density

9. What
is Vector?

10. Position of California from : orth Carolina is 3600《m

| N in

cal quantity

11. eae

nt from USA to China is (km)
/ en
physical quantity magnitude

direction

12. A vector is a physical quantity that has
both a magnitude and a direction.

Examples:
« Position ・ Acceleration
* Displacement + Momentum
* Velocity « Force

13. Representation of a vector
^ 多

も
Pe
VA =>
Tail

Direction

A

Symbolically it is represented as AB

14. Representation of a vector

They are also represented by a single capital
letter with an arrow above it.

a ×

15. Representation of a vector

Some vector quantities are represented by their
respective symbols with an arrow above it.
À

we

Position velocity Force

16. Types

“of Vectors
(on the basis of orient Hon)

17. Parallel Vectors

Two vectors are said to be parallel vectors, if

a they have same direction: "ES
LS

B 4
Re は 1 La
R= ao PU

= er - “St une HE)= A 3

18. Equal Vectors

Two parallel vectors are said to be equal vectors,
if they have same magnitude.

19. Anti-parallel Vectors

Two vectors are said to be anti-parallel vectors,
if they are in opposite directions.
pi > _
(なめ 一 一
a ツン —- +2

20. Negative Vectors

Two anti-parallel vectors are said to be negative
vectors, if they have same magnitude.

x
^

21. Collinear Vectors

Two vectors are said to be collinear vectors,
if they act along a same line.

Cra Qi
A Er

AO

22. Co-initial Vectors

Two or more vectors are said to be co-initial
vectors, if they have common initial point.

23. Co-terminus Vectors

Two or more vectors are said to be co-terminus
vectors, if they have common terminal point.

24. Coplanar Vectors

Three or more vectors are said to be coplanar
vectors, if they lie in the same plane.

ター

e
\

25. Non-coplanar Vectors

Three or more vectors are said to be non-coplanar
vectors, if they are distributed in space.

26. Types
of Vectors

(on the basis of éffect)

27. Polar Vectors

Vectors having straight line effect are
called polar vectors.

Examples:

ㆍ Displacement 。 Acceleration
+ Velocity * Force
> m

28. Axial Vectors

Vectors having rotational effect are
called axial vectors.

Examples:

・ Angular momentum + Angular acceleration
・ Angular velocity ・ Torque

29. Vector

Addition
(Geometrical Method)

30. Triangle Law

\

31. Parallelogram Law

32. Polygon Law

33. Commutative Property
で A

ー 2
A

C-A+B=B+A

Therefore, addition of vectors obey commutative law.

34. Associative Property

ol
の !

一

A

HERO)

Therefore, addition of vectors obey associative law.

35. Subtraction of vectors
> SN 4
B <> ,人
= ‘oO 7
A
The subtraction of 2 from vector A is defined as
the addition of vector -B to vector A.

CAE)

36. Vector
Addition

(Analytical Method)

37. Magnitude of Resultant

OC? = OA? + 20A x AM + AC?
In ACAM,

AM
cos 8 = > AM=AC 0050

In AOCM OC? = OA? + 20A x AC 0050
, 2
Oc? = OM? + CM? ae
R? = P? + 2P x Q cos 8 + 02
Oc? = (OA + AM)? + CM?

002 = 042 ar AM + AM? | R=./P2 + 2PQ cos 0 + 02 |

38. Direction of Resultant

sino = > CM=ACsind

AM
cos0= AC > AM=AC 0050

In AOCM,
CM
tana = OM
_ CM
tan 一 DATAM
AC sin 8
CK e FAC COS
tana = 250
P+Qcos@

39. Case I - Vectors are parallel (0 = 0°)

B E Q = R
Magnitude: Direction:
R = /P2 + 2PQ cos 0° + 02 eng ane
P+Qcos 0°
R = /P2 + 2PQ + @ 2
R = /@P+Q? tan =P+Q=0

40. Case II - Vectors are perpendicular (0 = 90°)
R E
u x |
P

Magnitude: Direction:
R = „/P2 + 2PQ cos 90° + 02 _ 99090" __Q
tana P+Qcos90° P+0
R = VP2 +0 + 02
a = tant Q
R= /P2+Q2 P

+ 0

B

41. Case III - Vectors are anti-parallel (8 = 180°)

P _

Magnitude:
R=/P2 + 2PQ cos 180° + Q2
R = /P2-2PQ+Q2

R= v(P—Q)?

le

Direction:

tana =

IfP > Q:

IfP < Q:

Q sin 180°

P+Q cos 180° ~

42. Unit vectors

A unit vector is a vector that has a magnitude of exactly
1 and drawn in the direction of given vector.

A

Az

+ It lacks both dimension and unit.
・ Its only purpose is to specify a direction in space.
|

43. ㆍ A given vector can be expressed as a product of
its magnitude and a unit vector.

。 For example A may be represented as,

A = magnitude of A
A = unit vector along A
|