Grade 12 STEM General Physics Dot and Cross Products of Vectors

Grade 12 STEM General Physics Dot and Cross Products of Vectors

1. Do t & Cr o ss pro d uct of vectors

2. What is dot product? o The dot product of two vectors A and B is defined as the scalar value AB cosθ , where θ is the angle between them such that 0≤θ≤π.

3. What is dot product? It is denoted by A . B by placing a dot sign between the vectors. So we have the equation, A . B = AB cosθ Another name of dot product is scalar product.

4. What is cross product? o The cross product of two vectors A and B is defined as AB sinθ with a direction perpendicular to A and B in right hand system, where θ is the angle between them such that 0≤θ≤π.

5. What is cross product? It is denoted by A x B by placing a cross sign between the vectors. So we have the equation, A x B = AB sinθ = C Another name of cross product is vector product.

6. History of dot product: Dot product was founded in 1901 in Vector Analysis by Edwin Bidwell Wilson: “ The direct product is denoted by writing the two vectors with a dot between them as A . B ” “ Thi s is re a d A dot B and ther e f o re m a y o f te n be called the dot product instead of the direct product ”

7. History of cross product: The first traceable work on ” cross product ” was founded in the book Vector Analysis . It was founded upon the lectures of Josiah Willard Gibbs , second edition by Edwin Bidwell Wilson published in 1909.

8. History of cross product: o On page 61, the mention of cross product was found for the first time. “ The skew product is denoted by a cross as the direct product was by a dot. It is written C = A x B and read A cross B . For this reason it is often called the cross product ” – Vector Analysis

9. Developing to present: While studying vector analysis, Gibbs noted that the product of quaternions always had to be separated into two parts: One dimensional quantity A three dimensional vector

10. Developing to present: To avoid this complexity he proposed defining distinct dot and cross products for pair of vectors and introduced the now common notation for them.

11. Confusion about representation: Dot product : Tait : Gibbs : Sαβ =Sβα α.β = β.α Cross product : Tait : Gibbs : Vαβ = –Vβα α x β = –β x α T o a v o id thi s representation complexit y , Gibbs’ notation is used universally.

12. Illustration of dot product: o Why Dot Produc t ? - To express the angular relationship between two vectors.

13. Illustration of dot product: If A and B are two vectors of form, A = A 1 i + A 2 j +A 3 k B = B 1 i + B 2 j + B 3 k Then the dot product of A and B is, A . B = A 1 B 1 + A 2 B 2 + A 3 B 3

14. Illustration of dot product: The angular relationship of two vectors A and B as per dot product is: A . B = │ A ││ B │ cosθ = AB cosθ

15. Illustration of dot product: The dot relationship of unit vectors along three axes : i . j = j . k = k . i = and i . i = j . j = k . k = 1

16. Illustration of cross product: o Why Cros s Product? - For accumulation of interactions between different dimensions.

17. Illustration of cross product: If A and B are two vectors of form A = A 1 i + A 2 j +A 3 k B = B 1 i + B 2 j + B 3 k Then the cross Product of A and B is, A x B = i j k A 1 A 2 A 3 B 1 B 2 B 3

18. Illustration of cross product: The angular relationship of two vectors A and B is A x B = │ A ││ B │ sinθ = AB sinθ

19. Illustration of cross product: The cross relationship of unit vectors along three axes are: i x i = j x j = k x k = i x j = k & j x i = -k j x k = i & k x j = -i k x i = j & i x k = -j

20. Properties of dot product: Co m mutative law: Distributive law: Associa t ive law: A . B = B . A A .( B + C ) = A . B + A . C m( A . B ) = (m A ) . B = A . (m B )

21. Properties of cross product: Distributive law: Associa t ive law: A x ( B + C ) = A x B + A x C m( A x B ) = (m A ) x B = A x (m B )

22. Distinction in commutative law: A x B = C ha s a magnitude ABsi n θ a n d direction is such that A , B and C form a right handed system (from fig-a ) θ A x B = C A B F i g - (a)

23. Distinction in commutative law: B x A = D has magnitude BAsin θ and direction such that B , A and D form a right handed system ( from fig -b )

24. Distinction in commutative law: Then D has the same magnitude as C but is opposite in direction, that is, C = - D A x B = - B x A Therefore the commutative law for cross product is not valid.

25. Applications of dot product: cos  = A . B AB AB  = co s − 1 ( A . B )  A ❶ Finding angle between two vectors: B A . B = | A || B | cos 

26. ❷ Projections of light: B A  Light source N O cos  = ON B ON = B cos  From the figure, cos  = A . B AB B cos  = A . B A As we know, ON = A . B A So we reach to,

27. Real life applications of dot product: Calculating total cost Electromagnetism, from which we get light, electricity, computers etc. Gives the combined effect of the coordinates in different dimensions on each other.

28. Applications of cross product: ❶ To find the area of a parallelogram: Area of parallelogram = h | B | = | A | sinθ | B | = | A x B |  A B h O C

29. Applications of cross product: Area of triangle = 1 2 h | B | = | A | sinθ | B | = 1 2 1 2 | A x B |  A ❷ To find the area of a triangle: B h O

30. Real life applications of cross product: Finding moment Finding torque Rowing a boat Finding the most effective path

31. Dot and cross vector together: Dot and cross products of three vectors A , B and C may produce meaningful products of the form ( A . B ) C , A . ( B x C ) and A x( B x C ) then phenomenon is called triple product. A .( B x C ) = A 1 A 2 A 3 B 1 B 2 B 3 C 1 C 2 C 3

32. Application of triple product: n A B h C Volume of the parallelepiped = (height h) x (area of the parallelogram I) = ( A .n) x (| B x C |) = A . (| B x C | n) = A . ( B x C ) I

33. Dot product vs cross product Dot product Cross product Result of a dot product is a scalar quantity. Result of a cross product is a vector quantity. It follows commutative law. It doesn’t follow commutative law. Dot product of vectors in the same direction is maximum. Cross product of vectors in same direction is zero. Dot product of orthogonal vectors is zero. Cross product of orthogonal vectors is maximum. It doesn’t follow right hand system. It follows right hand system. It is used to find projection of vectors. It is used to find a third vector. It is represented by a dot (.) It is represented by a cross (x)

34. Memory booster:  Area of a triangle of vectors is determined by which vector product method? A. Dot B. Cross

35. Memory booster:  Area of a triangle of vectors is determined by which vector product method? A. Dot B. Cross

36. Memory booster:  Area of a triangle of vectors is determined by which vector product method? A. Dot B. Cross  Projection of vectors is determined by which vector product method? A. Cross B. Dot

37. Memory booster:  Area of a triangle of vectors is determined by which vector product method? A. Dot B. Cross  Projection of vectors is determined by which vector product method? A. Cross B. Dot

38. Calculate |a × b| when |a| = 2, |b| = 4 and the angle between a and b is θ = 45◦ Calculate |a . b| when |a| = 2, |b| = 4 and the angle between a and b is θ = 90◦