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Mathematics

Mathematics. Session. Vectors -1. Session Objectives. Components of a Vector along and perpendicular to. Scalar or Dot Product. Geometrical Interpretation: Projection of a Vector. Properties of Scalar Product. Scalar Product in Terms of Components. Geometrical Problems.

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Mathematics

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  1. Mathematics

  2. Session Vectors -1

  3. Session Objectives • Components of a Vector along and perpendicular to • Scalar or Dot Product • Geometrical Interpretation: Projection of a Vector • Properties of Scalar Product • Scalar Product in Terms of Components • Geometrical Problems • Application: Work Done by a Force • Class Exercise

  4. Given two non-zero vectors inclined at an angle , the scalar product of , denoted by , is defined as Scalar or Dot Product Note: 1. The scalar product of two vectors is always a scalar quantity.

  5. Angle Between Two Vectors in Terms of Scalar Product

  6. Let be unit vectors along three mutually perpendicular coordinate axis, x-axis, y-axis and z-axis respectively. Orthonormal Vector Triad

  7. B M A O L Geometrical Interpretation of Scalar Product

  8. Projection of a Vector

  9. 1. Scalar product of two vectors is commutative 2. Scalar product of vectors is distributive Properties of Scalar Product

  10. 5. If m is a scalar and and be any two vectors, then 6. If and are two vectors, then Properties of Scalar Product (Cont.)

  11. Scalar Product in Terms of Components

  12. Components of a Vector along and perpendicular to

  13. Example -1

  14. Example –2

  15. Example –3

  16. Solution Cont. Solving (i), (ii) and (iii), we get x = –2, y = –3 and z = –4

  17. Example –4

  18. Example –5

  19. Solution Cont.

  20. Example –6

  21. Example –7

  22. Solution Cont.

  23. Geometrical ProblemsExample -8 Prove the cosine formula for a triangle, i.e. if a, b and c are the lengths of the opposite sides respectively to the angles A, B and C of a triangle ABC, then

  24. Solution By the triangle law of addition, we have

  25. Solution Cont.

  26. Solution Cont.

  27. Example -9 Show that the diagonals of a rhombus bisect each other at right-angles. Solution:

  28. Solution Cont. Hence, diagonals of a rhombus are at right angles.

  29. A(origin) D Example -10 Prove using vectors: The median to the base of an isosceles triangle is perpendicular to the base. Solution: Let ABC be an isosceles triangle in which AB = AC.

  30. Solution Cont.

  31. B A O Let be the displacement. Then the component of in the direction of the force is . Application: Work Done by a Force

  32. Example -11

  33. Example -12

  34. Solution Cont.

  35. Thank you

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