1. VECTORS
Wallin

2. Objectives and Essential Questions
Distinguish between basic trigonometric functions (SOH CAH TOA)
Distinguish between vector and scalar quantities
Add vectors using graphical and analytical methods
Essential Questions
What is a vector quantity? What is a scalar quantity? Give examples of each.

3. VECTOR
Recall: A VECTOR quantity is any quantity in physics that has BOTH MAGNITUDE and DIRECTION
Vector
Example
Magnitude and
Direction
Velocity
35 m/s, North
Acceleration
10 m/s2, South
Force
20 N, East
An arrow above the symbol
illustrates a vector quantity.
It indicates MAGNITUDE and
DIRECTION

4. Vector Properties 3 Basic Vector Properties:
Vectors can be moved, order of vectors doesn’t matter
Vectors can be added together
Vectors can be subtracted from one another

5. Moving Vectors
Vectors can be moved parallel to themselves in a diagram. The angle and magnitude of a vector cannot be changed.
This is often called the Tip to Tail Method.

6. Tip to Tail Method
When given two vectors, line up each vector tip to tail.
Example: Consider two vectors: one vector is 3 meters east and the other is 4 meters east.
Tip
3 meters east
4 meters east
Tail

7. Addition
ADDITION: When two (2) vectors point in the SAME direction, simply line them up
and add them together.
EXAMPLE: A man walks 46.5 m east, then another 20 m east.
Calculate his displacement relative to where he started. +
46.5 m, E
20 m, E
MAGNITUDE relates to the
size of the arrow and
DIRECTION relates to the
way the arrow is drawn
66.5 m, E

8. Subtraction
SUBTRACTION: When two (2) vectors point in the OPPOSITE directions,
simply line them up and subtract them.
EXAMPLE: A man walks 46.5 m east, then another 20 m west.
Calculate his displacement relative to where he started.
46.5 m, E -
20 m, W
26.5 m, E

9. Determining Vectors Graphically

10. Graphing Vectors.
Drawing vectors using a ruler and protractor to graphically represent vectors using arrows.
Rules:
Set a scale, Ex: 1 pace = 1 cm
Length of arrow indicates vector magnitude.
Use a protractor from the origin to find the angle, we call this angle θ.

11. Cartesian Coordinate System

12. Coordinate System
North
West
East
South

13. Determining direction
Option to report a direction
Report all angles from 0°
Report angle from the nearest axis
EX: 35°West of North OR °
What the heck does West of North mean??
The second direction, in this case North, is the direction that you begin facing.
The first direction, West in this case, is the direction that you turn.
N-face
W-turn E S

14. Graphing Vectors Example
Graphically resolve the following vector into its horizontal and vertical components
60 meters at 30º North of East

15. Graphical Resolution
Draw an accurate, to scale vector using a ruler and protractor.
Scale: 1 cm = 1 meter N
60.0 m
Vector goes a little bit north
30° W E
Vector goes a little east
We label these components Dx and Dy. S

16. Resolving Components
Use a ruler to measure the length of each component. N
Scale: 1 cm = 1 meter
60.0 m
Dy = 30.0 cm 30.0 meters
30° W E
Dx = 52.0 cm 52.0 meters
Your measurement will have slight variations, but should be very close because you drew your vector with care. S

17. Determining Vectors Mathematically

18. Math Review Function Abbrev. Description Sine sin opp. / hyp.
Cosine cos adj. / hyp.
Tangent tan opp. / adj.
Use inverse functions to find an angle when triangle side are known.
-Use the inverse button on your calculator.
Ex: tan -1, cos -1, sin -1

19. Mathematical Addition of Vectors at Angles
Sketch the vectors.
Break each vector into X & Y components using trig function
Put all X & Y components into a chart with appropriate signs.
Add all X & Y components
Redraw new triangle**.
**if necessary
Vector X Y 1 2 3
Total

20. Mathematical Addition of Vectors at Angles
Use Pythagorean Theorem to find the resultant.
Use inverse tangent to find the angle with respect to the coordinate system.
Write complete answer, including magnitude, unit, angle and direction.

21. 2-D VECTOR Example
Example: A man travels 120 km east then 160 km north. Calculate his resultant displacement.
FINISH
Vector
X (km)
Y (km) 1
120 2
160
Total
the hypotenuse is
called the RESULTANT
160 km, N
VERTICAL
COMPONENT
START
120 km, E
HORIZONTAL COMPONENT

22. 2-Dimensional VECTORS
When two (2) vectors are PERPENDICULAR to each other, you must
use the PYTHAGOREAN THEOREM
FINISH
160 km, N
VERTICAL
COMPONENT
START
120 km, E
HORIZONTAL COMPONENT

23. NEED A VALUE – ANGLE!
Just putting N of E is not good enough (how far north of east ?).
We need to find a numeric value for the direction.
To find the value of the angle we use a Trig function called TANGENT.
200 km
160 km, N
We call the angle theta q
120 km, E
So the COMPLETE final answer is : °North of East

24. Example
EX: A bear, searching for food wanders 35 meters east then 20 meters north. Frustrated, he wanders another 12 meters west then 6 meters south. Calculate the bear's displacement.
Vector
X (m)
Y (m) 1 35 2 20 3
-12 4 -6
Total 23 14
12 m, W
6 m, S
20 m, N
35 m, E R
14 m, N q
23 m, E
The Final Answer: m, 31.3 degrees NORTH of EAST

25. Example (pg 95)
A hiker walks 25.5 km from her base camp at 35° south of east. On the second day, she walks 41.0 km in a direction 65° north of east, at which point she discovers a forest ranger’s tower. Determine the magnitude and direction of her resultant displacement between the base camp and the ranger’s tower.

26. Consider a problem that asks you to find A – B.
Subtracting Vectors
Consider a problem that asks you to find A – B.
Approach:
A – B is the same thing as A + (–B)
To adjust the B vector add 180°to the measured angle, thereby “flipping” it
Example:
2 30° becomes °
Then continue vector addition as usual.