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 .

04/10/23 10.6: Vectors in Geometry

Vectors

Mathematical quantities with direction and magnitude (measure).

04/10/23 10.6: Vectors in Geometry

Examples of Vectors

Wind:

west at 25 miles per hour

04/10/23 10.6: Vectors in Geometry

Examples of Vectors

Gravity:

Down at 9.8 meters per second per second

04/10/23 10.6: Vectors in Geometry

Examples of Vectors

Pushing:

South with a force of 100N

04/10/23 10.6: Vectors in Geometry

Drawing Vectors

A

B

initial point

terminal point

shows direction

04/10/23 10.6: Vectors in Geometry

Naming Vectors

B

A

u

AB

u

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

04/10/23 10.6: Vectors in Geometry

Magnitude

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

04/10/23 10.6: Vectors in Geometry

Reference Vector

Positive x-axis

East

04/10/23 10.6: Vectors in Geometry

Equal Vectors

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

u

vu = v

04/10/23 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.

04/10/23 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.

04/10/23 10.6: Vectors in Geometry

Opposite Vectors

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

w

04/10/23 10.6: 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.

04/10/23 10.6: Vectors in Geometry

Methods for Addition of Vectors

Ordered Pairs

Head to Tail Method

Parallelogram Method

04/10/23 10.6: 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)

04/10/23 10.6: Vectors in Geometry

Head to Tail Method

Add AB + CD

A

DC

B

04/10/23 10.6: Vectors in Geometry

Head to Tail Method

Let C’D’ = CD

A

DC

B C’D’

04/10/23 10.6: Vectors in Geometry

Head to Tail Method

Translate C’D’ such that C’ = B

A

BC’ D’

04/10/23 10.6: Vectors in Geometry

Head to Tail Method

AD’ is the resultant vector

A

BC’ D’

04/10/23 10.6: 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.

04/10/23 10.6: Vectors in Geometry

5 mph

2 mph

04/10/23 10.6: Vectors in Geometry

Parallelogram Method for Adding Vectors

Add u + v

u

v

04/10/23 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.

04/10/23 10.6: 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.

04/10/23 10.6: 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’

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’

04/10/23 10.6: 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 Pluto’s actual movement.

04/10/23 10.6: Vectors in Geometry

Mic

key

Minnie

actual

04/10/23 10.6: 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?

04/10/23 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

E

S

W

04/10/23 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.”

0

90

180

270

04/10/23 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.”

0

90

180

270

04/10/23 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?

04/10/23 10.6: Vectors in Geometry

Assignment

pages 677- 679,

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