1. Vectors in the Plane and in Three-Dimensional Space

2. Points in Three-Dimensional Space y x z (2,3,4)

3. Distance Between Two Points

4. Open Ball ofof

5. Closed Ball ofof

6. Directed Line Segment A B

7. Opposite A B

8. Example

9. Definition of Vector

10. Example

11. Definition of Vector A vector is a quantity that is determined by a magnitude and a direction.

12. Speed vs. Velocity

13. Vector and Scalar Quantities Velocity Displacement Momentum Force Torque Acceleration Speed Distance Voltage Temperature Time Volume Mass

14. Computer Graphics Robot Arms

15. Cartesian Coordinate Represenatation of a Vector

16. Find the coordinate representation.

17. (3,2)

18. Cartesian Coordinate Represenatation of a Vector

19. Definition

20. Vector Addition y x z (2,3,4) (1,-2,2)

21. Vector Addition y x z (2,3,4) (1,-2,2) (3,1,6)

22. Vector Addition y x z (2,3,4) (1,-2,2) (3,1,6)

23. Vector Addition y x z (2,3,4) (1,-2,2) (3,1,6)

24. Vector Subtraction

28. Zero Vector

29. Scalar Multiplication

30. x y z

31. Length of a Vector

32. Special Unit Vectors

33. x y z i j k

34. Unit Vector Representation y x z

35. Direction of a Vector

36. Parallel Vectors

37. The Triangle Inequality

38. Properties

39. Application

40. Problems Show that the diagonals of a parallelogram bisect each other. Show that the line through the midpoints of adjacent sides of a parallelogram bisect one of the diagonals in the ratio 1:3. Show that the quadrilateral whose vertices are the midpoints of the sides of an arbitrary quadrilateral is a parallelogram.