Motion in 2 dimensions
3.1 -3.4
Vectors vs. Scalars
• Scalar- a quantity described by magnitude
only.
– Given by numbers and units only.
– Ex. Distance, speed and mass
• Vector – a quantity described by both
magnitude and direction
– Numbers, units and direction (either words or
angles
– Ex. Displacement, velocity, acceleration and
force
vectors
•
•
•
•
Drawn with an arrow
Length indicates magnitude
Direction pointed
Written in text either
35 m
East
– As a boldfaced letter
– Or as a letter with an arrow on top.
Vector subtraction
Scalar multiplication
• Vectors may be subtracted by adding the
opposite (or negative) of the second
vector.
V2 – V1 = V2 + (– V1)
– This means that V1 is facing the opposite
direction it was originally heading
• Multiplying a vector by a scalar simply
magnifies its length if V= 3 m then 3V = 9
m
Vector addition
• Vectors may be added two ways:
– One by graphing
• Vectors are drawn so that they remain in the same
orientation but are placed tip to tail.
• Or by the parallelogram method
– One by adding the components of each and
doing the Pythagorean theorem.
• The resultant vector is the outcome of
adding 2 vectors together.
Parallelogram method:
• In the parallelogram
method for vector
addition, the vectors are
translated, (i.e., moved)
to a common origin and
the parallelogram
constructed as follows:
• The resultant R is the
diagonal of the
parallelogram drawn from
the common origin.
• R = resultant vector
Pythagorean theorem
5 2 + 10 2 = R 2
R = 11.2 Km
Θ = tan -1 (5/10) =26.6°
west of north
So sum = 11.2 Km @
26.6° west of north
Component method – make a right
triangle out of the individual vector
components
• The tip of the x-component
vector is directly below the tip
of the original vector.
components
• .
• ,
When adding 2 vectors by the
component method
• Find the x and y components of each vector
• Organize in a table
• Then add x component to x component and y
component to y component,
Vector
X component
Y component
Xr
Yr
A
B
Resultant
• Use negative x components when vector
is pointed to the left
• Use Negative y components pointed down
• Use the Pythagorean formula to find the
length of the resulting vector.
• Find the angle using the Tan-1 (Yr /Xr)
• vector addition applet
• simulator
Example 1
• What is the vertical
component of a 33 m
vector that is at a 76°
angle with the x axis?
33
m
76°
• y –comp
• y = sin 76
33
y = 33 sin 76 = 32 m
Example 2
• A plane is heading to a
destination 1750 due
north at 175 km/hr in a
westward wind blowing
25 km/hr.
At what angle should the
plane be oriented so that it
reaches its destination?
Wind =25 km/hr
Plane
= 175
km/hr
• The wind will push the
plane off course to the
west by an angle of tan-1
(25/175) = 8° west of
north so the plane needs
to head 8° east of north.
Example 3
• Using the same plane in example 2, what
would the magnitude of the resultant
vector be?
– Since north and west, don’t need
components, just use the pythagorean
theorem.
1752 + 25 2 = R 2
R = 177 km/hr
example 4
• What vector
represents the
displacement of a
person who walks 15
km at 45° south of
east then 30 km due
west?
45°
15 km
30 km
•
•
•
•
Vector
X–
comp
Y -comp
15 km
10.6
-10.6
30 km
-30
0
resultant
-19.4
-10.6
R = √( -19.4) 2 + (-10.6)2
R = 22.1 km
Θ = tan -1 (-10.6/-19.4)
Θ = 29° south of west