Vectors and Scalars
Physics
Bell Ringer 10/13/15

Answer the following on your bell ringer
sheet:

1. Is displacement a vector or scalar?
2. What is the difference between
vectors & scalars?
NB: Don’t forget to write your objective
in your notebook before we start.
Scalar
A SCALAR is ANY
quantity in physics that
has MAGNITUDE, but
NOT a direction
associated with it.
Magnitude – A numerical
value
Scalar
Example
Magnitude
Speed
20 m/s
Distance
10 m
Age
15 years
Heat
1000
calories
Vector
A VECTOR is ANY
quantity in physics that
has BOTH
MAGNITUDE and
DIRECTION.
   
v , x, a, F
Vector
Velocity
Magnitude
& Direction
20 m/s, N
Acceleration 10 m/s/s, E
Force
5 N, West
Objective

We will use a treasure hunt activity to create
a vector addition map

I will map the course taken and add vectors
to find the resultant.

3F “ graphical vector addition”
Agenda



Cornell Notes- Essential Questions
Vector treasure hunt
Vector Map
Cornell Notes




What is a vector?
How do we represent vectors?
How do we draw a vector?





Essential Questions:
What shows the magnitude?
What shows the direction?
When should vectors be added?
When should vectors be subtracted?
What is the resultant vector?
Vectors
Vectors Quantities can be represented
with;
1.
2.
3.
Arrows
Signs (+ or -) 1-D motion
Angles and Definite Directions
(North, South, East, West)
Vectors

Every Vector

Tail
Head
Vectors are illustrated
by drawing an
ARROW above the
symbol. The head of
the arrow is used to
show the direction
and size of the arrow
shows the magnitude
Vector Addition
VECTOR ADDITION – If 2 similar vectors point in
the SAME direction, add them.

Example: A man walks 54.5 meters east, then another 30
meters east. Calculate his displacement relative to where he
started?
54.5 m, E
+
84.5 m, E
30 m, E
Vector Subtraction
VECTOR SUBTRACTION - If 2 vectors are going in
opposite directions, you SUBTRACT.

Example: A man walks 54.5 meters east, then 30
meters west. Calculate his displacement relative to
where he started?
54.5 m, E
30 m, W
24.5 m, E
-
Resultant

The vector representing the sum of two or
more vectors is called the resultant vector.
Vector Treasure Hunt





Create directions that lead to a specific
object/picture. You must have at least 4
turns.
Generate a map from your origin ( door) to
your picture.
Write each direction on an index card.
Scramble the index cards and follow them
again.
Then,( tomorrow) Create a vector map of
total displacement
Compass
Front of
room
Front door
Back of room
Teacher Model

Directions to object with at least 4 turns from
the front door.
Create a Map

Map drawn to scale.
Example
You walk 35 meters east then 20 meters north. You walk another 12 meters west
then 6 meters south. Calculate the Your displacement.
-
12 m, W
-
=
6 m, S
20 m, N
35 m, E
14 m, N
R
q
23 m, E
23 m, E
=
14 m, N
50 ft
East
Vector Map
25 ft
South
Expectation






Groups of 3-4
2 minutes to find the picture
10 minutes back track steps and create
directions on index cards
2 mins Shuffle cards
Try to locate object from directions.
10 mins draw map to scale
Group Pictures








Group1- gorilla
Group2- snake
Group 3- rat
Group 4- eagle
Group 5- alligator
Group 6- chicken
Group 7- pig
Group 8- dog
END
10/15/15 Bell Ringer ( 5 minutes!!)



Get back to your
groups.
1.On large paper,
create a title,
2. Create a map of
your directions from
origin to picture.


3. Create a Vector map
by Summing up the
vectors to find the
horizontal and vertical
components. Draw your
resultant.
Write your names & turn
in to teacher
Example
You walk 35 meters east then 20 meters north. You walk another 12 meters west
then 6 meters south. Calculate the Your displacement.
-
12 m, W
-
=
6 m, S
20 m, N
35 m, E
14 m, N
R
q
23 m, E
23 m, E
=
14 m, N
Bell Ringer 10/15/15 4 minutes


You walked 15 m east
from the door, then you
walked 6 m south, then
you turned around and
walked back west 2 m,
and another 6 meters west
then you walked North 7
m.
Draw a vector diagram to
show the motion and find
the resultant.
Lesson objectives

We will use Pythagorean theorem to find the
resultant, horizontal, and vertical components
of vectors.

I will resolve a vector into its vertical and
horizontal components and find the resultant
using Pythagorean theorem.
Agenda


Bell ringer
Problems 1-4




Diagrams only
Pythagorean theorem
( Solve for R)
Grade papers
Right angle triangles & Trig
functions

Note: You
Need a
calculator
today & your
notebook
Vector Addition Worksheet

Using Calculators




Make sure it is in
degrees
Locate the Sin, Cos,
& Tan buttons.
Locate the 2nd button.
Locate the Tan-1
Button.
Vectors

The hypotenuse in Physics
is called the RESULTANT.
A man walks 95 km, East then 55
km, north. Calculate his
RESULTANT DISPLACEMENT.
Finish
c2  a2  b2  c  a2  b2
55 km, N
Horizontal Component
95 km,E
Start
Vertical
Component
c  Resultant  952  552
c  12050  109.8 km
Vectors

The hypotenuse in Physics
is called the RESULTANT.
A man walks 95 km, East then 55
km, north. Calculate his
RESULTANT DISPLACEMENT.
Finish
c2  a2  b2  c  a2  b2
55 km, N
Horizontal Component
95 km,E
Start
Vertical
Component
c  Resultant  952  552
c  12050  109.8 km
Vectors

The hypotenuse in Physics
is called the RESULTANT.
A man walks 95 km, East then 55
km, north. Calculate his
RESULTANT DISPLACEMENT.
Finish
c2  a2  b2  c  a2  b2
55 km, N
Horizontal Component
95 km,E
Start
Vertical
Component
c  Resultant  952  552
c  12050  109.8 km
BUT……what about the direction?
In the previous example, DISPLACEMENT was asked for
and since it is a VECTOR we should include a
DIRECTION on our final answer.
N
W of N
E of N
N of E
N of W
NE
E
W
NOTE: When drawing a right triangle that
conveys some type of motion, you MUST
draw your components HEAD TO TOE.
S of W
S of E
W of S
E of S
S
Bell Ringer 10/16/15




Its Friday!!!!!
Check your grade in skyward. Write down
your current 2nd 6 weeks grade and your
semester grade average.
Turn in Bell Ringer sheet.
Remember: Tests must be made up within a
week of the test. Monday 10/19 is the
deadline. Also late policy is in effect!
Agenda

Review right angle triangle & Trig
functions ( SOH CAH TOA)



Example problems



How to enter Sin,Cos,Tan examples
How to enter Sin-1,Cos-1, Tan-1
examples
How to find missing sides.
How to find missing angles
Complete problems on your own

Note: You
Need a
calculator
today
Lesson objectives

We will use trigonometry to find the missing
side and the angles from right angle vector
diagrams.

I will find the missing angle and missing side
using trigonometry: Sine, Cosine, and
Tangent functions.
Using Calculators




Make sure it is in
degrees
Locate the Sin, Cos,
& Tan buttons.
Locate the 2nd button.
Locate the Tan-1
Button.
Using Calculators

Example 1: Use your
calculator to find the
sin, cos, or tan or any
angle.

Example 2: use your
calculator to find the
angle using the inverse
( Sin-1, Cos-1,& Tan-1)
Right angle
triangle, 90
Adjacent- Near the
angle
Hypotenuse- the
longest side
Opposite- opposite
the angle
What if you are missing a side?
Which do I use?
What if you are missing a side?
Which will you use to find the missing sides?
Pythagorean theorem or SOHCAH TOA?
Which do I use?
Trig
triangles
What if you are looking for the
angle?

To find the value of the angle we use a Trig
function called the Inverse.
Which sides do we have?
Which function do we use?
What if you are looking for the
angle?

To find the value of the angle we use a Trig
function called the Inverse.
What if you are looking for the angle?
Worksheet: Lesson 1
Reviewing the Primary Trigonometric
ratios
Examples
What if you are missing a component?
Suppose a person walked 65 m, 25 degrees East of North. What
were his horizontal and vertical components?
H.C. = ?
V.C = ?
25
65 m
The goal: ALWAYS MAKE A RIGHT
TRIANGLE!
To solve for components, we often use
the trig functions since and cosine.
adjacent side
hypotenuse
adj  hyp cos q
cosine q 
Let’s
identify
the sides
opposite side
hypotenuse
opp  hyp sin q
sine q 
adj  V .C.  65 cos 25  58.91m, N
SOH- CAH - TOA!!!!
opp  H .C.  65 sin 25  27.47m, E
What if you are missing a component?
Suppose a person walked 65 m, 25 degrees East of North. What
were his horizontal and vertical components?
H.C. = ?
V.C = ?
25
65 m
The goal: ALWAYS MAKE A RIGHT
TRIANGLE!
To solve for components, we often use
the trig functions since and cosine.
adjacent side
hypotenuse
adj  hyp cos q
cosine q 
opposite side
hypotenuse
opp  hyp sin q
sine q 
adj  V .C.  65 cos 25  58.91m, N
opp  H .C.  65 sin 25  27.47m, E
SOH- CAH - TOA!!!!
What if you are missing a component?
Suppose a person walked 65 m, 25 degrees East of North. What
were his horizontal and vertical components?
H.C. = ?
V.C = ?
25
65 m
The goal: ALWAYS MAKE A RIGHT
TRIANGLE!
To solve for components, we often use
the trig functions since and cosine.
adjacent side
hypotenuse
adj  hyp cos q
cosine q 
opposite side
hypotenuse
opp  hyp sin q
sine q 
adj  V .C.  65 cos 25  58.91m, N
opp  H .C.  65 sin 25  27.47m, E
BUT…..what about the VALUE of the
angle???
Just putting North of East on the answer is NOT specific enough
for the direction. We MUST find the VALUE of the angle.
To find the value of the
angle we use a Trig
function called the Inverse.
109.8 km
55 km, N
q N of E
95 km,E
opposite side 55
Tanq 

 0.5789
adjacent side 95
q  Tan 1 (0.5789)  30
So the COMPLETE final answer is : 109.8 km, 30 degrees North of East
BUT…..what about the VALUE of the
angle???
Just putting North of East on the answer is NOT specific enough
for the direction. We MUST find the VALUE of the angle.
To find the value of the
angle we use a Trig
function called the Inverse.
109.8 km
55 km, N
q N of E
95 km,E
opposite side 55
Tanq 

 0.5789
adjacent side 95
q  Tan 1 (0.5789)  30
109.8 km, 30 degrees North of East
BUT…..what about the VALUE of the
angle???
Just putting North of East on the answer is NOT specific enough
for the direction. We MUST find the VALUE of the angle.
.
109.8 km
55 km, N
q N of E
95 km,E
opposite side 55
Tanq 

 0.5789
adjacent side 95
q  Tan 1 (0.5789)  30
So the COMPLETE final answer is : 109.8 km, 30 degrees North of East
Example
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
20 m, N
35 m, E
q
=
14 m, N
R  14 2  232  26.93m
14 m, N
R
23 m, E
14
 .6087
23
q  Tan 1 (0.6087)  31.3
Tanq 
23 m, E
The Final Answer: 26.93 m, 31.3 degrees NORTH or EAST
Example
A boat moves with a velocity of 15 m/s, N in a river which
flows with a velocity of 8.0 m/s, west. Calculate the
boat's resultant velocity with respect to due north.
Rv  82  152  17 m / s
8.0 m/s, W
15 m/s, N
Rv
q
8
Tanq 
 0.5333
15
q  Tan 1 (0.5333)  28.1
The Final Answer : 17 m/s, @ 28.1 degrees West of North
Example
A plane moves with a velocity of 63.5 m/s at 32 degrees South of East. Calculate
the plane's horizontal and vertical velocity components.
adjacent side
cosine q 
hypotenuse
adj  hyp cos q
H.C. =?
32
63.5 m/s
opposite side
sine q 
hypotenuse
opp  hyp sin q
V.C. = ?
adj  H .C.  63.5 cos 32  53.85 m / s, E
opp  V .C.  63.5 sin 32  33.64 m / s, S
Example
A storm system moves 5000 km due east, then shifts course at 40
degrees North of East for 1500 km. Calculate the storm's
resultant displacement.
1500 km
adjacent side
hypotenuse
V.C.
adj  hyp cos q
cosine q 
opposite side
hypotenuse
opp  hyp sin q
sine q 
40
5000 km, E
H.C.
adj  H .C.  1500 cos 40  1149.1 km, E
opp  V .C.  1500 sin 40  964.2 km, N
5000 km + 1149.1 km = 6149.1 km
R  6149.12  964.2 2  6224.14 km
964.2
 0.157
6149.1
q  Tan 1 (0.364)  8.91
Tanq 
R
964.2 km
q
6149.1 km
The Final Answer: 6224.14 km @ 8.91
degrees, North of East
Hmm. That was good.
Holy Snakes!!!
It’s a bird, it’s a plane, It’s Super Rat!
Do I look like I’m playing?
Gator’s anyone?
Don’t be a chicken??
Do you think I’m cute? Yes or Nah? Yessss!!!!
What a time to be alive? 