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Understanding Vector Product: Properties, Applications, and Geometrical Representations

This session delves into the concept of the vector (cross) product, exploring its geometrical representation and key properties such as non-commutativity and distributivity over addition. Learn to express the vector product in terms of components and discover its applications, including calculating the moment of a force about a point or a line. The session will feature exercises on finding unit vectors perpendicular to planes, determining areas of parallelograms and triangles formed by vectors, and using vector methods to solve geometrical problems.

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Understanding Vector Product: Properties, Applications, and Geometrical Representations

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Presentation Transcript

  1. Mathematics

  2. Session Vectors -2

  3. Session Objectives • Vector (or Cross) Product • Geometrical Representation • Properties of Vector Product • Vector Product in Terms of Components • Applications: Vector Moment of a Force about a Point, about a Line • Class Exercise

  4. O Vector (or Cross) Product

  5. Note

  6. Geometrical Representation

  7. 1. Vector product is not commutative 2. Vector product is distributive over vector addition Properties of Vector Product

  8. Properties of Vector Product Cont.

  9. Properties of Vector Product Cont.

  10. Vector Product in Terms of Components

  11. Vectors Normal to the Plane of Two Given Vectors

  12. Lagrange’s Identity

  13. Find a unit vector perpendicular to the plane containing the vectors . Example –1

  14. Solution Cont.

  15. Solution: The vector perpendicular to the plane ABC is . Example –2

  16. Example –3

  17. Solution Cont.

  18. Example –4

  19. Solution Cont.

  20. Example –5

  21. Solution Cont.

  22. Applications

  23. Find the area of a parallelogram determined by the vectors Example -6

  24. Solution Cont.

  25. Example -7 Find the area of the triangle formed by the points A(1, 1, 2), B(2, 3, 5) and C(1, 5, 5).

  26. Solution Cont.

  27. O P Moment of Force About a Point

  28. Example -8

  29. The moment of the force acting through B about the point A is given by Solution Cont.

  30. Moment of Force About a Line

  31. Example -9

  32. The moment of the force about the given line is Solution Cont.

  33. In a triangle ABC, prove by vector method that: Geometrical Problem Example -10 Solution: By triangle law of vector addition

  34. Solution Cont.

  35. Solution Cont. From (i) and (ii), we get

  36. Thank you

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