PHYSICS STUDY GUIDE
CHAPTER 4: VECTORS
TOPICS:
• Vectors
WHAT YOU MUST KNOW
• Indentify vector physical quantities
• Label a vector
• Add two vectors
• Add multiple vectors
• Find the x-component and y-component of a vector at an angle
DEFINITIONS
•
•
•
I'm applying for a villain loan. I
go by Vector. It's a mathematical
term, represented by an arrow
with both magnitude and
direction. Vector! That's me,
because I commit crimes with
both direction and magnitude. Oh
yeah!
Vector: Physical quantity that requires magnitude and direction.
Resultant: Addition of two or more vectors
Vector component: decomposition of a vector resultant in a horizontal component (x-direction) and
a vertical component (y-direction)
VECTOR PHYSICAL QUANTITIES STUDIED IN CLASS (SO FAR)
•
•
•
•
•
dx
∆dx
vx
ax
Fx
or
or
or
or
or
dy
∆dy
vy
ay
Fy
Position
Displacement
velocity
Acceleration
Force
REPRESENTING VECTORS
•
Vectors are represented with arrows:
o The length of the arrow represents the magnitude (number) of the vector.
o The tip of the arrow represents the direction of the vector.
LABELING VECTORS
• Labeling vectors include:
o Symbol
o Direction
o Magnitude
o Units
EXAMPLES:
Fx = + 3 N
ADDING VECTORS:
Adding vectors means to find the vector resultant (see definitions above)
Vectors in the same direction: Add the vectors
Vectors in different directions: Use the Pythagorean theorem
•
•
•
EXAMPLE: Adding two vectors in different directions
1. Label your vectors
2. Think inside the box
Fx = + 8 N
•
Fy = - 8 N
Copy and paste each
vector: same
magnitude & same
direction.
3. Draw the vector
resultant from two
starting points
(together) to two
ending points
(together.
4. Find the resultant
using
the
Pythagorean
Theorem:
(F)2 = (Fx)2 + (Fy)2
(F)2 = (+8N)2 + (-8N)2
Fx
Fx
(F)2 = (64 N2) + (64 N2)
Fy
Fy
Fy
Fy
Fy
(F)2 = 128 N2
Fx
Fx
Fx
(F)2 = 128 N2
F = 11.31 N
FINDING THE COMPONENTS OF A VECTOR AT AN ANGLE
Each vector quantity at an angle has a:
•
•
Horizontal component also known as the x-component.
Vertical component also known as the y-component.
Look and find in the picture vector A.
Vector A is both horizontal and vertical, we say it is at an
angle.
The Greek letter θ (theta) is used to identify and angle in
the sketch.
θ (theta) must be measured always from the horizontal
line.
To find the vertical and horizontal components of vector A
you must use a scientific calculator.
Make sure you find the functions in your calculator
• SIN (SINE)
• COS (COSINE)
The y-component of the vector also known as the vertical component can be found as Ay = A · sin (θ).
The x-component of the vector also known as the horizontal component can be found as AX = A · cos (θ).
NOTE: Please make sure and double check that your calculator is set to work with angles in degrees. It
must display on the screen D or D or DEG or deg.
EXAMPLE: A rope is holding a sign at an angle of 65°. The tension force is measured to be 200 N. Find
the x-component and y-component of the tension force.
1. Your situation
2. Think inside the box
•
4. Find the x-component
your vector:
of
Copy and paste each vector:
same magnitude & same
direction.
5. Find the y-component
your vector:
FTx = FT · cos (θ)
FTY = FT · sin (θ)
FTx = 200 N · cos (65°)
FTY = 200 N · sin (65°)
FTx = 84.52 N
FTY = 181.26 N
The direction of this vector is
towards the left, therefore we
assign the negative direction.
FTx = - 84.52 N
3. Label the components of your
vector.
of
6. You may find the resultant using
the Pythagorean theorem
(FT)2 = (FTx)2 + (FTy)2
(FT)2 = (- 84.52 N)2 + (+ 181.26 N)2
(FT)2 = (7144.24 N2) + (32855.75 N2)
(FT)2 = 40000 N2
(FT )2 = 40000 N2
FT = 200 N
(we already knew that) !
ADDING MULTIPLE VECTORS:
•
•
•
•
•
•
•
Label each vector
Determine what vectors are along the x-direction
Determine what vectors are along the y-direction
Think inside the box for vectors at an angle
Add all vectors in the x-direction. This will be the x-direction of the resultant vector.
Add all vectors in the y-direction. This will be the y-direction of the resultant vector.
Find the resultant vector using the Pythagorean Theorem
1. Your situation: Six velocities are represented
below
2. Label each vector.
Vectors at an angle: Think inside the box
vy
Vx= +4 m/s
Vx= +3 m/s
Vy=+4 m/s
Vy=+4 m/s
Vy=-6 m/s
Vx=-6 m/s
Vx=+4 m/s
vx
3. Make a data table with the information of the xcomponent and y-component of each vector.
4. Draw
the
x-component
component of the resultant.
Find the sum of each component at the bottom of the table.
Find the magnitude of each component in
the data table
VECTOR
XCOMPONENT
Y
COMPONENT
V1
+ 4 m/s
0 m/s
V2
0 m/s
+ 4 m/s
+ 4 m/s
V3
+ 4 m/s
V4
+3 m/s
0 m/s
V5
0 m/s
- 6 m/s
V6
- 6 m/s
0 m/s
+ 5 m/s
+ 2 m/s
Σ
5. Find the resultant using the Pythagorean theorem
(v)2 = (vx)2 + (vy)2
(v)2 = (+5 m/s)2 + (+ 2 m/s)2
(v)2 = (25 m2/s2) + (4 m2/s2)
(v)2 = 29 m2/s2
(v)2 = 29 m2 /s2
v = 5.34 m/s
and
y-