1/16/201410.6: Vectors in Geometry Expectation: L1.2.3: Use vectors to represent quantities that...

04/10/23 10.6: Vectors in Geometry

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 .


Vectors

Mathematical quantities with direction and magnitude (measure).


Examples of Vectors

Wind:

west at 25 miles per hour


Examples of Vectors

Gravity:

Down at 9.8 meters per second per second


Examples of Vectors

Pushing:

South with a force of 100N


Drawing Vectors

A

B

initial point

terminal point

shows direction


Naming Vectors

B

A

u

AB

u

If A(0,0) and B(x,y), then u = (x,y)


Magnitude

We use the symbol | v | to denote the magnitude (measure) of vector v.


Reference Vector

Positive x-axis

East


Equal Vectors

Defn: Two vectors are equal iff they have the same direction and magnitude.

u

vu = v


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.


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.


Opposite Vectors

Two vectors are opposite vectors iff their magnitudes are equal, but their directions are opposite. v

w


Addition of Vectors

-combination of forces

-sum of 2 vectors is called the resultant vector.

ex: two people pushing on the same object.


Methods for Addition of Vectors

Ordered Pairs

Head to Tail Method

Parallelogram Method


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)


Head to Tail Method

Add AB + CD

A

DC

B


Head to Tail Method

Let C’D’ = CD

A

DC

B C’D’


Head to Tail Method

Translate C’D’ such that C’ = B

A

BC’ D’


Head to Tail Method

AD’ is the resultant vector

A

BC’ D’


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.


5 mph

2 mph


Parallelogram Method for Adding Vectors

Add u + v

u

v



Add u + v

u

v’

Let v’ = v. Translate v’ to the initial point of u.


Add u + v

v’’


u

v’

Let v’’ = v. Translate v’’ to the terminal point of u.


Add u + v

v’’


u

v’

Let u’ = u. Translate u’ to the terminal point of v’.

u’

10.6: Vectors in Geometry

Add u + v

v’’


u

v’

The sum is the vector along the diagonal of the parallelogram.

u’


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.


Mic

key

Minnie

actual


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?


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


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.”

0

90

180

270


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?


Assignment

pages 677- 679,

#11-23 (odds), 24-26 (all)

1/16/201410.6: Vectors in Geometry Expectation: L1.2.3: Use vectors to represent quantities that...

Documents

Transcript of 1/16/201410.6: Vectors in Geometry Expectation: L1.2.3: Use vectors to represent quantities that...