1. Vectors in a Plane Mesa Verde National Park, Colorado
Photo by Vickie Kelly, 2003
Greg Kelly, Hanford High School, Richland, Washington

2. The length is written as
In the past we’ve only worked with lines that have slopes but not necessarily direction.
Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments.
There’s a difference between going 50 mph north and 50 mph south B
The length is written as
terminal
point A
initial
point
Look, it’s just the distance formula

3. The component form of this vector is:
In the past we’ve only worked with lines that have slopes but not necessarily direction.
Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments.
There’s a difference between going 50 mph north and 50 mph south
The component form of this vector is: B
terminal
point
i marks the x-component of the vector A
initial
point
j marks the y-component of the vector

4. The component form of this vector is:
In the past we’ve only worked with lines that have slopes but not necessarily direction.
Quantities such as force, displacement or velocity that have direction as well as magnitude are represented by directed line segments.
There’s a difference between going 50 mph north and 50 mph south
The component form of this vector is: B
terminal
point A or
initial
point
Notice here that we always subtract the initial point from the terminal point because we need to establish direction

5. Find the component form of each vector
(–3, 4) P
(2, 2) B
(–5, 2) Q A

6. Notice that the vectors are pointed in opposite directions
Find the component form of each vector
(–3, 4) P
(2, 2) B
(–5, 2) Q A
Notice that the vectors are pointed in opposite directions
but have the same length.

7. Vector Addition: u + v u + v is the resultant vector. u v yv xv u+v vyu u xu
Clean this up and add examples
(Add the components.)

8. u – v u – v u v (Subtract the components.) v – v u – v u
MC #1 pg 62, #1 pg 63, #3 pg 64

9. What if we only knew the length of the vector and the angle?
A vector is in standard position if the initial point is at the origin. x
What if we only knew the length of the vector and the angle?
The component form of this vector is: or
Before anyone panics, this is just SOHCAHTOA…
Just watch…

10. Remember what this really means:
A vector is in standard position if the initial point is at the origin.
y component x
x component
x component
Remember what this really means:
Think of it as a hypotenuse of the right triangle above because it’s the length of the vector or
x component
y component
x component
y component
See? Just SOHCAHTOA

11. If it’s the angle that you need to find, then you need to know this:
The direction of a vector u is found this way:
This is just the x component divided by the magnitude
The sign follows the sign of the y component

12. Unit Vectors
Unit Vectors, just like r in the unit circle, have a length of 1. The formula for finding any given unit vector is simple:
The unit vector for is
Now we’re going to try a few
homework problems…

13. Application:
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E

14. Application:
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E u

15. Application:
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N v
60o E u

16. Application:
A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N
We need to find the magnitude and direction of the resultant vector u + v. v
u+v E u

17. The component forms of u and v are:70
u+v E
500 u
Therefore:
and:

18. The component forms of u and v are:
u+v E
Therefore:
Remembering that
You can also use tan-1

19. N
538.4
6.5o E
The new ground speed of the airplane is about mph, and its new direction is about 6.5o north of east.