VECTORS
Wallin
Objectives and Essential
Questions
 Objectives
 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.
VECTOR
Recall: A VECTOR quantity is any
quantity in physics that has BOTH
MAGNITUDE and DIRECTION
 
x ,v ,a ,F
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
Vector Properties
3 Basic Vector Properties:
1. Vectors can be moved, order of vectors doesn’t
matter
2. Vectors can be added together
3. Vectors can be subtracted from one another
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.
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
Tail
4 meters east
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
+
66.5 m, E
20 m, E
MAGNITUDE relates to the
size of the arrow and
DIRECTION relates to the
way the arrow is drawn
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
Determining Vectors
Graphically
Graphing Vectors.
Drawing vectors using a ruler and
protractor to graphically represent vectors
using arrows.
Rules:
1. Set a scale, Ex: 1 pace = 1 cm
2. Length of arrow indicates vector
magnitude.
3. Use a protractor from the origin to find
the angle, we call this angle θ.
Cartesian Coordinate System
Coordinate System
North
West
East
South
Determining direction
Option to report a direction
1. Report all angles from 0°
2. Report angle from the nearest axis
EX: 35°West of North
OR
125°
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
Graphing Vectors Example
Graphically resolve the following
vector into its horizontal and vertical
components
60 meters at 30º North of East
Graphical Resolution
Draw an accurate, to scale vector using a ruler
and protractor.
Scale: 1 cm = 1 meter
N
Vector goes a little bit north
30°
W
Vector goes a little east
S
E
We label these components Dx and Dy.
Resolving Components
Use a ruler to measure the length of each
component.
N
Scale: 1 cm = 1 meter
Dy = 30.0 cm 30.0 meters
30°
W
Dx = 52.0 cm 52.0 meters
S
E
Your measurement will have slight variations, but
should be very close because you drew your vector
with care.
Determining Vectors
Mathematically
Math Review
Function
Abbrev.
Description
Sine
Cosine
Tangent
sin
cos
tan
opp. / hyp.
adj. / hyp.
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
Mathematical Addition of
Vectors at Angles
1. Sketch the vectors.
2. Break each vector into X & Y components using trig
function
3. Put all X & Y components into a chart with appropriate
signs.
4. Add all X & Y components
5. Redraw new triangle**.
**if necessary
Vector
1
2
3
Total
X
Y
Mathematical Addition of
Vectors at Angles
6. Use Pythagorean Theorem to find the resultant.
7. Use inverse tangent to find the angle with respect to the
coordinate system.
8. Write complete answer, including magnitude, unit,
angle and direction.
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
0
the hypotenuse is
called the RESULTANT
160 km, N
2
0
160
Total
120
160
VERTICAL
COMPONENT
S
R
T
T
A
120 km, E
HORIZONTAL COMPONENT
2-Dimensional VECTORS
When two (2) vectors are PERPENDICULAR to each other, you must
use the PYTHAGOREAN THEOREM
c a b
2
2
2
 c a b
2
FINISH
2
2
2

c  resultant  120   160  


c  200 km
S
R
T
T
A
160 km, N
VERTICAL
COMPONENT
120 km, E
HORIZONTAL COMPONENT
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
Tan 
We call the
angle theta

opposite side 160

 1.333
adjacent side 120
  Tan1 (1.333)  53.1o
120 km, E
So the COMPLETE final answer
is : 200 km @ 53.1 °North of East
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.
12 m, W
6 m, S
Vector
X (m)
Y (m)
1
35
0
2
0
20
3
-12
0
4
0
-6
Total
23
14
20 m, N
35 m, E
14 m, N
R

23 m, E
R  14 2  232  26.93m
14
 .6087
23
  Tan 1 (0.6087)  31.3
Tan 
The Final Answer: 26.93 m, 31.3 degrees NORTH of EAST
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.
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 meters @ 30° becomes
2 meters @ 210°
 Then continue vector addition as usual.