1. Introduction to Vectors UNIT 1

2. What is a Vector? A vector is a directed line segment, can be represented as AB or AB, where A and B are the two endpoints of the line segment. Directed means that the vector has a direction. QUESTION: Which direction is implied for vector AB?

3. Vectors: Example A B

4. Vector Quantity There are exactly two properties that completely characterize a vector: 1)Direction – which way does the vector point? 2)Magnitude – the length of the vector, written as |XY| for vector XY Together the direction and magnitude define the vector quantity. QUESTION: What are some examples of vectors that we are already familiar with?

5. Equal Vectors Two vectors are equal vectors if they have both the same direction and magnitude

6. Three cars on the road are driving in the same direction at the same speed: Do they have equal velocity vectors?

7. Opposite Vectors: An opposite vector is a vector with the same magnitude as the original, but opposite direction: A B In this illustration, AB and BA are opposite vectors.

8. Vector Sum A vector sum A + B is defined as a vector that results from placing the initial point of vector B at the terminal point of vector A: the vector with the same initial point as A and the same terminal point as B is the vector sum. A B A + B

9. Parallelogram Rule Another method, called the parallelogram rule, to find A + B is to place vectors A and B so that their initial points coincide, then complete a parallelogram that has A and B as two adjacent sides. The diagonal of the parallelogram with the same initial points is the vector sum:

10. Parallelogram Rule Example A B A + B

11. The two methods side by side: A B A + B A B The two methods give identical results for the vector sum

12. QUESTION Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force: 22 15 100°

13. QUESTION Two forces of 15 newtons and 22 newtons act at a point in the plane. If the angle between the forces is 100°, find the magnitude of the resultant force: 22 15 =80 Z 22 100°

14. Scalar Product A real number k and a vector U create the vector kU which has the magnitude |k| times the magnitude of U. kU has the same direction as U if k>0, and the opposite direction if k < 0: – So vector 2U would be twice the length of U and would point in the same direction as U does:

15. Scalar Product Examples U 2U 3U -U -2U -3U

16. Position Vectors A position vector is a vector with its initial point at the origin and with its endpoint at (a, b). It is written, so U = below: (a, b) U X axis Y axis (0, 0)

17. Direction Angle of The direction angle is the positive angle between the x-axis and a position vector: (a, b) U X axis Y axis (0, 0) Direction Angle

18. Direction Angle of (cont) (a, b) U X axis Y axis (0, 0) The direction angle θ satisfies tan θ = b/a, where a ≠ 0: b a

19. Magnitude of vector (a, b) U X axis Y axis (0, 0) b a

20. QUESTION What is the direction angle and magnitude of vector U= ? (3, -2) U X axis Y axis (0, 0)

21. QUESTION

22. Horizontal and Vertical Components of a Vector The horizontal and vertical components of a vector U are given by: a = |U|cos θ b = |U|sin θ (a, b) U X axis Y axis (0, 0) b = |U|sin θ a = |U|cos θ

23. QUESTION Calculate the vertical and horizontal components of a vector with direction angle of 40° and a magnitude of 25. (a, b) |U|=25 X axis Y axis (0, 0) = 40

24. QUESTION Calculate the horizontal and vertical components of a vector with direction angle of 40° and a magnitude of 25. ANSWER: x = 19.2, y = 16.1 or

25. Vector Operations Overview + = k* = If A =, then –A = - = + - OR + =

26. Vector Operations For any real numbers a, b, c, and d: + = (a, b) X axis Y axis (0, 0) (c, d) (a+c, b+d)

27. Vector Operations For any real numbers a, b, c, and d: + = (3, 4) X axis Y axis (0, 0) (4, -2) (7, 2)

28. Vector Operations (cont) B If U = Then –U =

29. Vector Operations Scalar multiplication: k* = Examples: -3* = 6* = 0* =

30. Vector Operations (0, 0) (3, 4) (0, 0) (6, 8) U 2U

31. QUESTION Consider the vectors shown in the following figure, and perform the operations: U V U + V (4, 3) X axis Y axis (0, 0) (-2, 1) (x, y) a)U + V b)-2U c)4U – 3V

32. Vector Subtraction A B -A -B Vector subtraction is the inverse operation of vector addition and is defined as adding the negative vector: So we have B – A = B + (-A) for all vectors A, B Therefore (see below) B – A = C C

33. Vector Subtraction - = + - OR + = Examples: - =

34. Vector Subtraction QUESTIONS QUESTION: Express A as a difference of two vectors. QUESTION: Express B as a difference of two vectors. A B C

35. Vector Notation Conventions Unit Vectors: i =, j= i, j Form for Vectors: If v =, then v = ai + bj

36. Unit Vectors (5, 3) U X axis Y axis (0, 0) i i i i i j j j i j U= 5i + 3j

37. QUESTION Write the vector in the form ai + bj:

38. QUESTION Write the vector in the form ai + bj: Answer: If v=, then: V=ai + bj, so V = -5i + 8j

39. Dot Product The dot product of the two vectors U = and V = is denoted by UV, read “U dot V,” is given by UV = ac + bd. Examples: =

40. Geometric Interpretation of the Dot Product If θ is the angle between two nonzero vectors U and V, where 0° < θ < 180°, then UV = |U|*|V|cos θ Example: = 6 using =ac+bd But it is also true using UV = |U|*|V|cos θ

41. Geometric Interpretation of the Dot Product (3, 3) U X axis Y axis (0, 0) (2, 0) V θ=45

42. Properties of the Dot Product UV = VU U(V+W)=UV + UW (U + V)W = UW + VW (kU)V = k(UV) = U(kV) 0U = 0 UU = |U| 2

43. The Dot Product Can Be Positive, Zero, or Negative θ θ < 90: Positive dot product θ θ θ > 90: Negative dot product θ = 90: Zero dot product