1. Vector-Valued Functions 12 Copyright © Cengage Learning. All rights reserved.

2. Vectors in the Plane Copyright © Cengage Learning. All rights reserved. 11.1 12.1 Vector-Valued Functions

3. Component Form of a Vector

4. Many quantities in geometry and physics, such as area, volume, temperature, mass, and time, can be characterized by a single real number scaled to appropriate units of measure. These are called scalar quantities, and the real number associated with each is called a scalar. Other quantities, such as force, velocity, and acceleration, involve both magnitude and direction and cannot be characterized completely by a single real number.

5. So, quantities we measure that have magnitude but not direction are called scalars. Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments. A B initial point terminal point The length is

6. A B initial point terminal point A vector is represented by a directed line segment. Vectors are equal if they have the same length and direction (same slope).

7. A vector is in standard position if the initial point is at the origin. x y The component form of this vector is:

8. A vector is in standard position if the initial point is at the origin. x y The component form of this vector is: The magnitude (length) of is:

9. P Q (-3,4) (-5,2) The component form of is: v (-2,-2)

10. If Then v is a unit vector. is the zero vector and has no direction.

11. In Theorem 11.3, u is called a unit vector in the direction of v. The process of multiplying v by to get a unit vector is called normalization of v. Unit Vectors

12. Vector Operations

13. (Add the components.) (Subtract the components.) Vector Operations

14. Scalar Multiplication: Negative (opposite): Vector Operations

15. v v u u u+v u + v is the resultant vector. (Parallelogram law of addition)

16. Generally, the length of the sum of two vectors is not equal to the sum of their lengths. To see this, consider the vectors u and v as shown in Figure 11.9. Figure 11.9 Vector Operations

17. By considering u and v as two sides of a triangle, you can see that the length of the third side is ||u + v||, and you have ||u + v|| ≤ ||u|| + ||v||. Equality occurs only if the vectors u and v have the same direction. This result is called the triangle inequality for vectors. Vector Operations

18. In surveying and navigation, a bearing is a direction that measures the acute angle that a path or line of sight makes with a fixed north-south line. In air navigation, bearings are measured in degrees clockwise from north. Applications of Vectors

19. Application: A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E

20. Application: A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E u

21. Application: A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u 60 o

22. Application: A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u We need to find the magnitude and direction of the resultant vector u + v. u+v

23. N E v u The component forms of u and v are: u+v 500 70 Therefore: and:

24. N E The new ground speed of the airplane is about 538.4 mph, and its new direction is about 6.5 o north of east. How would this new direction be stated in air bearings? 538.4 6.5 o

25. Application – Finding the Resultant Force Two tugboats are pushing an ocean liner, as shown below. Each boat is exerting a force of 400 pounds. What is the resultant force on the ocean liner?

26. Solution Using Figure 11.12, you can represent the forces exerted by the first and second tugboats as F 1 = = F 2 = =

27. The resultant force on the ocean liner is F = F 1 + F 2 = = So, the resultant force on the ocean liner is approximately 752 pounds in the direction of the positive x-axis. Solution

28. The dot product (also called inner product) is defined as: Read “u dot v” Example: Vector Operations

29. The dot product (also called inner product) is defined as: This could be substituted in the formula for the angle between vectors (or solved for theta) to give: Vector Operations

30. Find the angle between vectors u and v : Example:

31. Standard Unit Vectors

32. The unit vectors and are called the standard unit vectors in the plane and are denoted by as shown in Figure 11.10. Any vector can be represented as a linear combination of these standard unit vectors.

33. Vector-Valued Functions Copyright © Cengage Learning. All rights reserved. 12.1

34. Space Curves and Vector-Valued Functions

35. Any vector can be written as a linear combination of two standard unit vectors. The vector v is a linear combination of the vectors i and j. The scalar a is the horizontal component of v and the scalar b is the vertical component of v.

36. We can describe the position of a moving particle by a vector, r ( t ). If we separate r ( t ) into horizontal and vertical components, we can express r ( t ) as a linear combination of standard unit vectors i and j.

37. In three dimensions the component form becomes:

38. A Space Curves and Vector-Valued Functions

39. With vector-valued functions, the terminal point of the position vector r(t) coincides with the point (x, y) or (x, y, z) on the curve given by the parametric equations, as shown below. Figure 12.1 Space Curves and Vector-Valued Functions

40. The arrowhead on the curve indicates the curve’s orientation by pointing in the direction of increasing values of t. Unless stated otherwise, the domain of a vector-valued function r is considered to be the intersection of the domains of the component functions f, g, and h. For instance, the domain of is the interval (0, 1]. Space Curves and Vector-Valued Functions

41. Sketch the plane curve represented by the vector-valued function r(t) = 2cos t i – 3sin t j, 0 ≤ t ≤ 2 . Vector-valued function Solution: From the position vector r(t), you can write the parametric equations x = 2cos t and y = –3sin t. Solving for cos t and sin t and using the identity cos 2 t + sin 2 t = 1 produces Rectangular equation Example 1 – Sketching a Plane Curve

42. The graph of this rectangular equation is the ellipse shown in Figure 12.2. The curve has a clockwise orientation. That is, as t increases from 0 to 2 , the position vector r(t) moves clockwise, and its terminal point traces the ellipse. Figure 12.2 cont’d Example 1 – Solution

43. Graph on the TI-89 using the parametric mode. MODE Graph…….2 ENTER Y= ENTER WINDOW GRAPH

44. Graph on the TI-89 using the parametric mode. MODE Graph…….2 ENTER Y= ENTER WINDOW GRAPH

45. Limits and Continuity

47. If r(t) approaches the vector L as t → a, the length of the vector r(t) – L approaches 0. That is, ||r(t) – L|| → 0 as t → a. This is illustrated graphically in Figure 12.6. Limits and Continuity

49. Discuss the continuity of the vector-valued function given by r(t) = ti + aj + (a 2 – t 2 )k at t = 0. a is a constant. Solution: As t approaches 0, the limit is Example 5 – Continuity of Vector-Valued Functions

50. Because r(0) = (0)i + (a)j + (a 2 )k = aj + a 2 k you can conclude that r is continuous at t = 0. By similar reasoning, you can conclude that the vector-valued function r is continuous at all real- number values of t. Example 5 – Solution cont’d

51. Homework: Section 11.1 MMM pgs. 173-174

52. Homework Section 12.1 pg. 837, #1-9 odd, 13, 17-35 odd, 45-51 odd