Understanding the Vector Dot Product: Definitions, Properties, and Applications

1. Vector Refresher Part 3 Vector Dot Product Definitions Some Properties The Angle Between 2 Vectors Scalar Projections Vector Projections

2. Dot Product • One form of vector multiplication • Yields a SCALAR quantity • Can be used to find the angle between 2 vectors • Can also be used to find the projection of a vector in a given direction

3. Symbolism • The dot product is symbolized with a dot between 2 vectors

4. Symbolism • The dot product is symbolized with a dot between 2 vectors • The following means “Vector A dotted with vector B”

5. One Definition The dot product is defined as the sum of the product of similar components of a vector

6. One Definition The dot product is defined as the sum of the product of similar components of a vector If we have the following 2 vectors:

7. One Definition The dot product is defined as the sum of the product of similar components of a vector If we have the following 2 vectors:

8. One Definition The dot product is defined as the sum of the product of similar components of a vector If we have the following 2 vectors: NOTE: This is a SCALAR term whose units are the product of the units of the 2 vectors

9. Another Definition The dot product is also related to the angle produced by arranging 2 vectors tail totail.

10. Another Definition The dot product is also related to the angle produced by arranging 2 vectors tail to tail. If we have the following 2 vectors: θ

11. Properties of the Dot Product Commutative:

12. Properties of the Dot Product Commutative: Associative:

13. Properties of the Dot Product Commutative: Associative: Distributive:

14. The Angle Between 2 Vectors The dot product is a useful tool in determining the angle between 2 vectors θ

15. The Angle Between 2 Vectors The dot product is a useful tool in determining the angle between 2 vectors θ

16. The Angle Between 2 Vectors The dot product is a useful tool in determining the angle between 2 vectors θ

17. The Angle Between 2 Vectors The dot product is a useful tool in determining the angle between 2 vectors θ

18. The Angle Between 2 Vectors The dot product is a useful tool in determining the angle between 2 vectors θ If 2 vectors are orthogonal, their dot product is 0

19. Scalar Projection The dot product is also used to determine how much of a vector is acting in a particular direction.

20. Scalar Projection The dot product is also used to determine how much of a vector is acting in a particular direction. θ

21. Scalar Projection The dot product is also used to determine how much of a vector is acting in a particular direction. If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product θ

22. Scalar Projection The dot product is also used to determine how much of a vector is acting in a particular direction. If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product θ

23. Scalar Projection The dot product is also used to determine how much of a vector is acting in a particular direction. If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product Note that this result is a SCALARquantity, meaning that it has no direction associated. θ

24. Scalar Projection The dot product is also used to determine how much of a vector is acting in a particular direction. If we want to find how much of acts in the direction of , (length of the green line) we can use the dot product Note that this result is a SCALARquantity, meaning that it has no direction associated. Thus, this calculation is the scalar projection θ

25. Vector Projection The scalar projection can be used to determine a vector projection We can transform the scalar projection, in this case , into a vector by multiplying the scalar projection and the unit vector that described the direction of interest, in this case θ This is a VECTOR quantity that describes the vector shown by the green arrow

26. Applications of the Vector Projection We can use the vector projection to determine the vector parallel and perpendicular to a given direction θ

27. Applications of the Vector Projection We can use the vector projection to determine the vector parallel and perpendicular to a given direction A vector can be described as its vector component parallel to a direction plus its component perpendicular to a direction θ

28. Applications of the Vector Projection We can use the vector projection to determine the vector parallel and perpendicular to a given direction A vector can be described as its vector component parallel to a direction plus its component perpendicular to a direction θ

29. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to .

30. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Looking at this formula, we need to determine the magnitude of each vector and evaluate the dot product

31. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can start by finding the magnitude of vector U

32. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can start by finding the magnitude of vector U

33. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can start by finding the magnitude of vector U

34. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can do the same for vector V

35. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can do the same for vector V

36. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can do the same for vector V

37. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Next, we’ll take the dot product to complete the formula.

38. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can use the inverse cosine function to find the angle

39. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can use the inverse cosine function to find the angle

40. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the projection of U onto V, we need to use the formula to the left, which means we need the unit vector that describes the direction of V

41. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We already calculated the magnitude of V. We’ll use that to find the unit vector

42. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We already calculated the magnitude of V. We’ll use that to find the unit vector

43. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can take the dot product to find the scalar projection.

44. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Now, we can take the dot product to find the scalar projection.

45. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the vector projection, we’ll apply the scalar projection to the unit vector that describes the direction of V.

46. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the vector projection, we’ll apply the scalar projection to the unit vector that describes the direction of V.

47. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . To find the vector projection, we’ll apply the scalar projection to the unit vector that describes the direction of V.

48. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Finally, we can subtract the component of U parallel to V from U to get the part of U that is perpendicular to V.

49. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . Finally, we can subtract the component of U parallel to V from U to get the part of U that is perpendicular to V.

50. Example Problem If and , find the angle between the 2 vectors, the projection of onto , and the component of that is perpendicular to . We can check our work with the following formula because the parallel and perpendicular components of U form a right triangle, with U as the hypotenuse.