1. Chapter 13 Vector Applications Contents:
A. Problems involving vector operations
B. Lines in 2-D and 3-D
C. The angle between two lines
D. Constant velocity problems
E. The shortest distance from a line to a point
F. Intersecting lines
G. Relationships between lines

3. A. PROBLEMS INVOLVING VECTOR OPERATIONS
The sum of vectors is called the resultant vector.

4. Example 1
In still water, Jacques can swim at 1.5m/s. Jacques is at point A on the edge of a canal, and considers point B directly opposite. A current is flowing from the left at a constant speed of 0.5m/s. a. If Jacques dives in straight towards B, and swims without allowing for the current, what will his actual speed and direction be? b. Jacques wants to swim directly across the canal to point B. i. At what angle should Jacques aim to swim in order that the current will correct his direction? ii What will Jacques’ actual speed be?

5. Suppose c is the current’s velocity vector, s is the velocity vector Jacques would have if the water was still, and f = c + s is Jacques’ resultant velocity vector. a. Jacques aims directly across the river, but the current takes him downstream to the right.

6. Applying the Pythagorean Theorem and a little trigonometry
Applying the Pythagorean Theorem and a little trigonometry. Jacques has an actual speed of approximately 1.58m/s and his direction of motion is approximately 18.4o to the right of his intended line.

7. b. Jacques needs to aim to the left of B so the current will correct his direction.

8. Exercise 13A
1. An athlete can normally run with constant speed 6 m/s. Using a vector diagram to illustrate each situation, find the athlete’s speed if: a. he is assisted by a wind of 1 m/s from directly behind him b. he runs into a head wind of 1 m/s.

9. a. b.
6 m/s
1 m/s
7 m/s
6 m/s
1 m/s
5 m/s

10. 5 An airplane needs to fly due east from one city to another at a speed of 400 km/h. However, a 50 km/h wind blows constantly from the north-east. a. How does the wind affect the speed of the airplane? b. In what direction must the airplane head to compensate for the wind?

11. a. Using the Law of Cosine, compute the value for x The calm airspeed of the plane 437 km/h which is slowed to 400 km/h by the northeasterly wind. x
135o
50 km/h
400 km/h

12. b. Determine q: use the Law of Sine. north of due east. x q
135o
50 km/h
400 km/h

13. LINES IN 2-D AND 3-D
In both 2-D and 3-D geometry we can determine the equation of a line using its direction and any fixed point on the line. Suppose a line passes through a fixed point A with position vector a, and that the line is parallel to the vector b.

15. Vector Equation of a line

16. A line passes through the point A(1, 5) and has direction vector
A line passes through the point A(1, 5) and has direction vector . Describe the line using: a. A vector equation, b. Parametric equations, c. and a Cartesian equation.

18. Parametric equation

19. Cartesian equation: Solve both equation for t
Cartesian equation: Solve both equation for t. Set them equal to one another and simplify.

20. LINES IN 3-D

21. Find a vector equation and the corresponding parametric equations of the line through (1,-2, 3) in the direction 4i + 5j – 6k.

23. NON-UNIQUENESS OF THE VECTOR EQUATION OF A LINE
Going from (5, 4) to (7, 3), go
2 to the right and 1 down.
Going from (7, 3) to (5,4), go
2 to the left and 1 up.

24. Find parametric equations of the line through A(2,-1, 4) and B(-1, 0, 2).

25. Find the directional vector AB or BA
Using point A
Using point B

26. Exercise 13B

33. Solve for t, set equal and simplify
Sub in (k, 4)

34. THE ANGLE BETWEEN TWO LINES

38. Exercise 13C
Find the angle between the lines L1: x = t, y = 3 + 5t L2: x = 3s, y = -6 – 4s

43. Find the measure of the acute angle between the lines 2x + y = 5 and 3x – 2y = 8.

45. Find the measure of the angle between the lines: y = 2 – x and x – 2y = 7 Answer: 71.6o

46. CONSTANT VELOCITY PROBLEMS

49. a. The initial position of the object occurs at t = 0

52. velocity vector
Is the velocity vector

55. To find the position of the object at time t, we need to find the parametric equation
for x and y. The coordinates of (x, y) will give us our position.
Step 1: set up the vector equation of the object.
Step 2: find the parametric equation from the vector equation
Step 3: write the coordinate as an ordered pair of parametric equations.

56. b. To determine the speed of the object, we need to find the magnitude or length
of the vector.
c. Plug in the value for t into our ordered pair.
d. Due east of (0, 0) means when y = 0 (aka x-intercept)

57. 22. EXERCISE 13d

58. 1a. The initial position occurs at t = 0
1a. The initial position occurs at t = 0. x(0) = 1 + 2(0) = 1 y(0) = 2 – 5(0) = 2 the initial position (1, 2)

59. 1b.
x = 1 + 2t
y = 2 - 5t
(x, y) 1 2
(1, 2) 3 -3
(3, -3) 5 -8
(5, -8) 7
-13
(7, -13)

63. THE SHORTEST DISTANCE FROM A LINE TO A POINT

65. Perpendicular/orthogonal

70. EXERCISE 13E

80. INTERSECTING LINES

83. EXERCISE 13F

87. RELATIONSHIPS BETWEEN LINES

88. LINE CLASSIFICATION IN 3 DIMENSION