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1/16/201410.6: Vectors in Geometry Expectation: L1.2.3: Use vectors to represent quantities that have magnitude and direction, interpret direction and.

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Presentation on theme: "1/16/201410.6: Vectors in Geometry Expectation: L1.2.3: Use vectors to represent quantities that have magnitude and direction, interpret direction and."— Presentation transcript:

1 1/16/ : 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.

2 1/16/ : Vectors in Geometry Vectors Mathematical quantities with direction and magnitude (measure).

3 1/16/ : Vectors in Geometry Examples of Vectors Wind: west at 25 miles per hour

4 1/16/ : Vectors in Geometry Examples of Vectors Gravity: Down at 9.8 meters per second per second

5 1/16/ : Vectors in Geometry Examples of Vectors Pushing: South with a force of 100N

6 1/16/ : Vectors in Geometry Drawing Vectors A B initial point terminal point shows direction

7 1/16/ : Vectors in Geometry Naming Vectors B A u AB u If A(0,0) and B(x,y), then u = (x,y)

8 1/16/ : Vectors in Geometry Magnitude We use the symbol | v | to denote the magnitude (measure) of vector v.

9 1/16/ : Vectors in Geometry Reference Vector Positive x-axis East

10 1/16/ : Vectors in Geometry Equal Vectors Defn: Two vectors are equal iff they have the same direction and magnitude. u v u = v

11 1/16/ : Vectors in Geometry 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.

12 1/16/ : Vectors in Geometry 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.

13 1/16/ : Vectors in Geometry Opposite Vectors Two vectors are opposite vectors iff their magnitudes are equal, but their directions are opposite.

14 1/16/ : Vectors in Geometry Addition of Vectors -combination of forces -sum of 2 vectors is called the resultant vector. ex: two people pushing on the same object.

15 1/16/ : Vectors in Geometry Methods for Addition of Vectors Ordered Pairs Head to Tail Method Parallelogram Method

16 1/16/ : Vectors in Geometry Ordered Pairs Method (a,b) + (c,d) = (a+c, b+d) u + v = (2,20) Find u + v if u = (4,8) and v = (-2,12)

17 1/16/ : Vectors in Geometry Head to Tail Method Add AB + CD A D C B

18 1/16/ : Vectors in Geometry Head to Tail Method Let CD = CD A D C B C D

19 1/16/ : Vectors in Geometry Head to Tail Method Translate CD such that C = B A B C D

20 1/16/ : Vectors in Geometry Head to Tail Method AD is the resultant vector A B C D

21 1/16/ : Vectors in Geometry 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.

22 1/16/ : Vectors in Geometry 5 mph 2 mph

23 1/16/ : Vectors in Geometry Parallelogram Method for Adding Vectors Add u + v u v

24 1/16/ : Vectors in Geometry Parallelogram Method for Adding Vectors Add u + v u v Let v = v. Translate v to the initial point of u.

25 1/16/ : Vectors in Geometry Add u + v v Parallelogram Method for Adding Vectors u v Let v = v. Translate v to the terminal point of u.

26 1/16/ : Vectors in Geometry Add u + v v Parallelogram Method for Adding Vectors u v Let u = u. Translate u to the terminal point of v. u

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

28 1/16/ : Vectors in Geometry 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 Plutos actual movement.

29 1/16/ : Vectors in Geometry Mickey Minnie actual

30 1/16/ : Vectors in Geometry 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?

31 1/16/ : Vectors in Geometry 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 E S W

32 1/16/ : Vectors in Geometry 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

33 1/16/ : Vectors in Geometry 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

34 1/16/ : Vectors in Geometry 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?

35 1/16/ : Vectors in Geometry Assignment pages , #11-23 (odds), (all)


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