1. 10.6: Vectors in Geometry Expectation:
L1.2.3: Use vectors to represent quantities that have magnitude and direction, interpret direction and magnitude of a vector numerically, and calculate the sum and difference of two vectors.
G1.3.2: Know and use the Law of Sines and the Law of Cosines and use them to solve problems. Find the area of a triangle with sides a and b and included angle θ using the formula Area = (1/2) a b sin θ .
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10.6: Vectors in Geometry

2. Vectors
Mathematical quantities with direction and magnitude (measure).
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10.6: Vectors in Geometry

3. Examples of Vectors Wind: west at 25 miles per hour 3/25/2017
10.6: Vectors in Geometry

4. Examples of Vectors Gravity: Down at 9.8 meters per second per second
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10.6: Vectors in Geometry

5. Examples of Vectors Pushing: South with a force of 100N 3/25/2017
10.6: Vectors in Geometry

6. Drawing Vectors shows direction B terminal point A initial point
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10.6: Vectors in Geometry

7. Naming Vectors B u A AB u If A(0,0) and B(x,y), then u = (x,y)
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10.6: Vectors in Geometry

8. Magnitude
We use the symbol | v | to denote the magnitude (measure) of vector v.
7th hour 4/10/06
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10.6: Vectors in Geometry

9. Reference Vector Positive x-axis East 3/25/2017
10.6: Vectors in Geometry

10. Equal Vectors
Defn: Two vectors are equal iff they have the same direction and magnitude. u v
u = v
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10.6: Vectors in Geometry

11. Parallel Vectors
Defn: Two vectors are parallel iff they have the same direction.
Ex: The wind is blowing from the west at 10 mph with gusts to 20 mph.
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10.6: Vectors in Geometry

12. Perpendicular Vectors
Defn: Two vectors are perpendicular iff their directions are at right angles to each other.
A plane is flying north at 200 mph and the wind is blowing from the east at 25 mph.
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10.6: Vectors in Geometry

13. Opposite Vectors
Two vectors are opposite vectors iff their magnitudes are equal, but their directions are opposite.
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10.6: Vectors in Geometry

14. Addition of Vectors -combination of forces
ex: two people pushing on the same object.
-sum of 2 vectors is called the resultant vector.
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10.6: Vectors in Geometry

15. Methods for Addition of Vectors
Ordered Pairs
Head to Tail Method
Parallelogram Method
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10.6: Vectors in Geometry

16. Ordered Pairs Method u + v = (2,20) (a,b) + (c,d) = (a+c, b+d)
Find u + v if u = (4,8) and v = (-2,12)
u + v = (2,20)
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10.6: Vectors in Geometry

17. Head to Tail Method Add AB + CD B C D A 3/25/2017
3rd hour 4/29/05 D A
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10.6: Vectors in Geometry

18. Head to Tail Method Let C’D’ = CD B C’ D’ C D A 3/25/2017
10.6: Vectors in Geometry

19. Head to Tail Method Translate C’D’ such that C’ = B B C’ D’ A
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10.6: Vectors in Geometry

20. Head to Tail Method AD’ is the resultant vector B C’ D’ A 3/25/2017
10.6: Vectors in Geometry

21. A rowboat is traveling due east at 5 mph
A rowboat is traveling due east at 5 mph. The current is pushing the boat due south at 2 mph. Show the direction the boat will actually travel.
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10.6: Vectors in Geometry

22. 5 mph
2 mph
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10.6: Vectors in Geometry

23. Parallelogram Method for Adding Vectors
Add u + v u v
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10.6: Vectors in Geometry

24. Parallelogram Method for Adding Vectors
Add u + v u v’
Let v’ = v. Translate v’ to the initial point of u.
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10.6: Vectors in Geometry

25. Parallelogram Method for Adding Vectors
Add u + v
v’’
Let v’’ = v. Translate v’’ to the terminal point of u. u v’
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10.6: Vectors in Geometry

26. Parallelogram Method for Adding Vectors
Add u + v
v’’ u u’ v’
Let u’ = u. Translate u’ to the terminal point of v’.
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10.6: Vectors in Geometry

27. Parallelogram Method for Adding Vectors
Add u + v
v’’ u u’ v’
The sum is the vector along the diagonal of the parallelogram.
10.6: Vectors in Geometry

28. Mickey and Minnie are each pushing Pluto towards his bath
Mickey and Minnie are each pushing Pluto towards his bath. If Mickey pushes north with a force of 5 N and Minnie pushes east with a force of 7N, draw the vector representing Pluto’s actual movement.
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10.6: Vectors in Geometry

29. Minnie
Mickey
actual
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10.6: Vectors in Geometry

30. A plane is flying due west at 150 mph
A plane is flying due west at 150 mph. The wind is pushing the plane 20° south of west at 18 mph. What are the actual speed and direction of the plane?
3rd hour 4/25/06
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10.6: Vectors in Geometry

31. If a direction is just given in terms of an angle measure, such as a heading of 175°, we need to use a “compass rose.” N W E S
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10.6: Vectors in Geometry

32. If a direction is just given in terms of an angle measure, such as a heading of 175°, we need to use a “compass rose.”
270 90
180
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10.6: Vectors in Geometry

33. If a direction is just given in terms of an angle measure, such as a heading of 175°, we need to use a “compass rose.”
270 90
180
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10.6: Vectors in Geometry

34. A boat needs to travel at a heading of 35°, but the current has a speed of 10 miles per hour from 165°. If the boats speed in still water is 25 miles per hour, at what heading should the boat travel to reach the 35° heading?
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10.6: Vectors in Geometry

35. Assignment pages 677- 679, #11-23 (odds), 24-26 (all) 3/25/2017
10.6: Vectors in Geometry