scaler and vectors

1. TEACHING ACADEMY SCALARS & VECTORS

2. TEACHING ACADEMY OUTLINE • SCALER • VECTOR • GEOMETRICAL AND GTAPHICAL REPRESENTATION • PROPERTIES • TYPES OF VECTORS

3. TEACHING ACADEMY PHYSICAL QUANTITIES • The quantities that we can measure is called physical quantities. • They have two types • Scalar Quantities 2. Vector quantities

4. What isScalar? TEACHING ACADEMY

5. TEACHING ACADEMY Length of a car is 4.5m physical quantity magnitude

6. TEACHING ACADEMY Mass of gold bar is 1kg physical quantity magnitude

7. TEACHING ACADEMY Time is 12.76s physical quantity magnitude

8. TEACHING ACADEMY Temperature is 36.8°C physical quantity magnitude

9. TEACHING ACADEMY Scalar is a physical quantity that haveonly magnitude • Examples: • Mass • Length • Time • Temperature • Volume • Density

10. TEACHING ACADEMY What is a vector

11. TEACHING ACADEMY North Carolina California Position of California from North Carolina is 3600km in west physical quantity magnitude direction

12. TEACHING ACADEMY USA China Displacement from USA to China is 11600km ineast physical quantity magnitude direction

13. TEACHING ACADEMY A vector is a physical quantity that has both magnitude and direction Examples: • Position • Displacement • Velocity • Acceleration • Momentum • Force

14. Representation of avector TEACHING ACADEMY Symbolically it is represented asAB

15. Representation of avector TEACHING ACADEMY They are also represented by a singlecapital letter with an arrow aboveit. A P B

16. Representation of avector TEACHING ACADEMY Some vector quantities are represented by their respective symbols with an arrow aboveit. r v F Position velocity Force

17. TEACHING ACADEMY Types ofVectors (on the basis oforientation)

18. TEACHING ACADEMY ParallelVectors Two vectors are said to be parallel vectors, if they have samedirection. P A Q B

19. EqualVectors Two parallel vectors are said to be equal vectors, if they have samemagnitude. P A B Q P =Q A =B

20. Anti-parallelVectors TEACHING ACADEMY Two vectors are said to be anti-parallel vectors, if they are in oppositedirections. P A Q B

21. NegativeVectors Two anti-parallel vectors are said to be negative vectors, if they have samemagnitude and opposite direction. P A B Q P =−Q A =−B

22. CollinearVectors Two vectors are said to be collinearvectors, if they act along a sameline. B Q A P

23. Co-initialVectors Two or more vectors are said to be co-initial vectors, if they have common initialpoint. B A C D

24. Co-terminusVectors Two or more vectors are said to be co-terminus vectors, if they have common terminalpoint. B A C D

25. CoplanarVectors Three or more vectors are said to be coplanar vectors, if they lie in the sameplane. D B

26. Non-coplanarVectors Three or more vectors are said to be non-coplanar vectors, if they are distributed inspace. B C A

27. Types ofVectors (on the basis ofeffect)

28. PolarVectors Vectors having straight line effect are called polar vectors. TEACHING ACADEMY Examples: • Displacement • Velocity • Acceleration • Force

29. AxialVectors TEACHING ACADEMY Vectors having rotational effect are called axialvectors. • Example • Angularmomentum • Angularvelocity • Angularacceleration • Torque

30. Vector Addition (GeometricalMethod)

31. TriangleLaw C B A B A C = A +B

32. ParallelogramLaw A B C A B A B C = A +B A B

33. PolygonLaw A B A D C E B D E = A + B + C +D C

34. CommutativeProperty A C B B C A C = A + B = B +A Therefore, addition of vectors obey commutativelaw.

35. AssociativeProperty C C D D B B A A D = (A + B) + C = A + (B +C) Therefore, addition of vectors obey associativelaw.

36. Subtraction of vectors A B −B A The subtraction of B from vector A is defined as the addition of vector −B to vectorA. A - B = A +(−B)

37. Vector Addition (AnalyticalMethod)

38. Magnitude ofResultant B C OC2 = OA2 + 2OA × AM +AC2 In∆CAM, Q R θ θ AM cos θ= ⇒ AM = AC cosθ O A M AC P In∆OCM, OC2 = OM2 +CM2 OC2 = (OA + AM)2 +CM2 OC2 = OA2 + 2OA × AM +AM2 + CM2 OC2 = OA2 + 2OA × AC cosθ +AC2 R2 = P2 + 2P × Q cos θ +Q2 R= P2+2PQcosθ+Q2

39. Direction ofResultant C B In∆OCM, Q R CM tanα = OM CM α P θ O A M tanα= OA+AM AC sinθ In∆CAM, tanα= OA+AC cosθ CM sin θ= ⇒ CM = AC sinθ AC Q sinθ tanα = P+Qcosθ AM cos θ= ⇒ AM = AC cosθ AC

40. Case I – Vectors are parallel (𝛉 =𝟎°) + = Q P R Magnitude: Direction: R= P2+2PQcos0°+Q2 R= P2 + 2PQ +Q2 R= (P +Q)2 Q sin0° tanα= P+Qcos0° 0 tanα = =0 P +Q R = P +Q α =0°

41. Case II – Vectors are perpendicular (𝛉 =𝟗𝟎°) R α + = Q Q P P Magnitude: R= P2+2PQcos90°+Q2 R= P2 + 0 +Q2 R= P2 +Q2 Direction: Q Q sin90° tanα= = P + Qcos90° P +0 Q P α =tan−1

42. Case III – Vectors are anti-parallel (𝛉 =𝟏𝟖𝟎°) − = Q P R Magnitude: Direction: R= P2+2PQcos180°+Q2 R= P2 − 2PQ +Q2 R= (P −Q)2 Q sin180° =0 tanα= P+Qcos180° α =0° If P >Q: If P <Q: α =180° R = P −Q

43. Unitvectors • A unit vector is a vector that has a magnitude of exactly 1 and drawn in the direction of givenvector. • A • 𝐴 • It lacks both dimension andunit. • Its only purpose is to specify a direction inspace.

44. Unitvectors • A given vector can be expressed as a product of its magnitude and a unitvector. • For example A may be representedas, • A = A𝐴 • A = magnitude ofA • 𝐴=unitvector alongA

45. Cartesian unitvectors 𝑦 −𝑧 𝑗 -𝑘 -𝑖 𝑘 𝑥 𝑖 −𝑥 -𝑗 𝑧 −𝑦

46. Resolution of aVector It is the process of splitting a vector into two or more vectors in such a way that their combined effect is same as that of the givenvector. 𝑛 A A𝑛 𝑡 A𝑡

47. Rectangular Components of 2DVectors 𝑦 A A𝑦𝑗 θ A A𝑦 A𝑥𝑖 θ 𝑥 O A = A𝑥𝑖 + A𝑦𝑗 A𝑥

48. Rectangular Components of 2DVectors A𝑦 ⇒ A𝑦 = A sinθ sin θ= A A A 𝑦 θ A𝑥 ⇒ A𝑥 = A cosθ cos θ= A A𝑥

49. Magnitude & direction fromcomponents Magnitude: A= A2 +A2 𝑥 𝑦 Direction: A = A𝑥𝑖 + A𝑦𝑗 A A𝑦 A𝑦 θ A𝑥 θ =tan−1 A𝑥

50. Rectangular Components of 3DVectors 𝑦 A = A′ +A𝑦 A A = A𝑥 + A𝑧 +A𝑦 A𝑦 A = A𝑥 + A𝑦 +A𝑧 A𝑥 𝑥 A𝑧 A′ A = A𝑥𝑖 + A𝑦𝑗 + A𝑧𝑘 𝑧