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 . 3/25/2017 10.6: Vectors in Geometry

Vectors Mathematical quantities with direction and magnitude (measure). 3/25/2017 10.6: Vectors in Geometry

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

Examples of Vectors Gravity: Down at 9.8 meters per second per second 3/25/2017 10.6: Vectors in Geometry

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

Drawing Vectors shows direction B terminal point A initial point 3/25/2017 10.6: Vectors in Geometry

Naming Vectors B u A AB u If A(0,0) and B(x,y), then u = (x,y) 3/25/2017 10.6: Vectors in Geometry

Magnitude We use the symbol | v | to denote the magnitude (measure) of vector v. 7th hour 4/10/06 3/25/2017 10.6: Vectors in Geometry

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

Equal Vectors Defn: Two vectors are equal iff they have the same direction and magnitude. u v u = v 3/25/2017 10.6: 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. 3/25/2017 10.6: 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. 3/25/2017 10.6: Vectors in Geometry

Opposite Vectors Two vectors are opposite vectors iff their magnitudes are equal, but their directions are opposite. 3/25/2017 10.6: Vectors in Geometry

Addition of Vectors -combination of forces ex: two people pushing on the same object. -sum of 2 vectors is called the resultant vector. 3/25/2017 10.6: Vectors in Geometry

Methods for Addition of Vectors Ordered Pairs Head to Tail Method Parallelogram Method 3/25/2017 10.6: Vectors in Geometry

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) 3/25/2017 10.6: Vectors in Geometry

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

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

Head to Tail Method Translate C’D’ such that C’ = B B C’ D’ A 3/25/2017 10.6: Vectors in Geometry

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

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. 3/25/2017 10.6: Vectors in Geometry

5 mph 2 mph 3/25/2017 10.6: Vectors in Geometry

Parallelogram Method for Adding Vectors Add u + v u v 3/25/2017 10.6: Vectors in Geometry

Parallelogram Method for Adding Vectors Add u + v u v’ Let v’ = v. Translate v’ to the initial point of u. 3/25/2017 10.6: Vectors in Geometry

Parallelogram Method for Adding Vectors Add u + v v’’ Let v’’ = v. Translate v’’ to the terminal point of u. u v’ 3/25/2017 10.6: Vectors in Geometry

Parallelogram Method for Adding Vectors Add u + v v’’ u u’ v’ Let u’ = u. Translate u’ to the terminal point of v’. 3/25/2017 10.6: Vectors in Geometry

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

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. 3/25/2017 10.6: Vectors in Geometry

Minnie Mickey actual 3/25/2017 10.6: Vectors in Geometry

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 3/25/2017 10.6: 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 W E S 3/25/2017 10.6: 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.” 270 90 180 3/25/2017 10.6: 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.” 270 90 180 3/25/2017 10.6: 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? 3/25/2017 10.6: Vectors in Geometry

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