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)