VECTORS AND TWODIMENSIONAL MOTION
Properties of Vectors
Vectors vs. Scalars
• Physical quantities can be categorized as:
Scalar quantities
Magnitude (with
appropriate units)
or
Vector quantities
Magnitude (with
appropriate units)
+
Direction
Examples:
Temperature
Mass
Time
Examples:
Velocity
Displacement
Acceleration
Adding Vectors graphically
• If we can add scalars then we can add vectors
 They must have the same units
m
m
m
 They must be connected from tip to tail
 Resultant (R)= A + B
TRIANGLE METHOD OF ADDITION
 Commutative Law of Addition
SUBTRACTING VECTORS
The negative of a Vector
A-B
ADDITION VS. SUBTRACTION
ADDITION
A
+
R
SUBTRACTION
A
- B
R
=0
EXAMPLE PROBLEM (We solve)
• GOAL: Find the sum of two vectors
A car travels 20.0 km due north and then 35.0 km in
a direction 60° west of north. Find the
magnitude and direction of the resultant vector.
This vector is called the car’s resultant
displacement (measure).
EXAMPLE PROBLEM (We solve)
Vector A has a magnitude of 29 units and
points in the positive y-direction. When
vector B is added to A, the resultant vector A +
B points in the negative y-direction with a
magnitude of 14 units. Find the magnitude
and direction of B.
COMPONENTS OF A VECTOR
A
AY
θ
AX
EXAMPLE PROBLEM
• Goal: Find vector components, given a
magnitude and direction
Find the horizontal and vertical
components of the 1.00x10^2 m
displacement of a superhero
who flies from the top of a tall
building along the path shown.
30.0°
100m
PRACTICE PROBLEM
• Goal: Find the resultant vector,
given its components
Suppose instead the superhero leaps in
the other direction along a
displacement vector B to the top of a
flagpole where the displacement
components are given by Bx= -25.0m
and By = 10.0m. Find the magnitude
and direction of the displacement
vector.
Practice Problem
Suppose the superhero had flown 150m
at a 120° angle with respect to the
positive x-axis. Find the components
of the displacement vector.
Practice Problem
Suppose instead the superhero had
leaped with a displacement having an xcomponent of 32.5 m and a y-component
of 24.3 m. Find the magnitude and
direction of the displacement vector.
ADDING VECTORS ALGEBRAICALLY
x-components with x-components only
Rx = Ax + Bx
y-components with y-components only
Ry = Ay + By
Subtracting vectors is the same thing,
because subtracting is the same as adding
the negative of one vector to the other.
PRACTICE PROBLEM
• Goal: Add vectors algebraically
and find the resultant vector
A hiker begins a trip by first walking 25.0
m 45.0° sourth of east from her base
camp. On the second day she walks
40.0 km in a direction 60.0° north of
east, at which point she discovers a
forest ranger’s tower.
a)
b)
c)
AY
Determine the components of the hiker’s
displacements in the first and second days.
Determine the components of the hiker’s total
displacement for the trip
Find the magnitude and direction of the
displacement from base camp
R
B
45.0°
A
60.0°
PRACTICE PROBLEM
A cruise ship leaving port travels 50.0 km 45.0°
north of west and then 70.0 km at a heading
30.0° north of east. Find:
a) The ship’s displacement vector and
b) The displacement vector’s magnitude and
direction.
EXAMPLE PROBLEM (You solve)
Vector A has a magnitude of 8.00 units and
makes an angle of 45.0° with the positive xaxis. Vector B also has a magnitude of 8.00
units and is directed along the negative x-axis.
Using graphical methods, find
(a) the vector sum A + B.
(b) The vector difference A - B
EXAMPLE PROBLEM (You+Partner solve)
Vector A is 3.00 units in length and points along
the positive x-axis. Vector B is 4.00 units in
length and points along the negative y-axis.
Use graphical methods to find the magnitude
and direction of the vectors
a) A + B
b) A - B
HOMEWORK
3 easy problems + one challenging problem
Posted on website.
Homework problem 1
Each of the displacement vectos A and B shown
has a magnitude of 3.00m. Find
a) A + B
b) A – B
c) B – A
3.00m
d) A – 2B
30°
0
Homework Problem 2
A roller coaster moves 200 ft horizontally and
then rises 135 ft at an angle of 30.0° above
the horizontal. Next, it travels 135 ft at an
angle of 40.0° below the horizontal. Find the
roller coaster’s displacement from its starting
point to the end of this movement.
Homework Problem 3
A plane flies from base camp to lake A, a
distance of 280 km at a direction of 20.0°
north of east. After dropping off supplies, the
plane flies to lake B, which is 190 km and 30.0°
west of north from lake A. Graphically
determine the distance and direction from
lake B to the base camp.
Homework Problem 4
An airplane flies 200 km due west from city A to
city B and then 300 km in the direction of
30.0° north of west from city B to city C.
a) In straight-line distance, how far is city C
from city A?
b) Relative to city A, in what direction is city C?
Homework Problem 5
A jogger runs 100 m due west, then changes
direction for the second leg of the run. At the
end of the run, she is 175 m away from the
starting point at an angle of 15.0° north of
west. What were the direction and length of
her second displacement?
Homework Problem 6
A man lost in a maze makes three consecutive
displacements so that at the end of his travel
he is right back where he started. The first
displacement is 8.00 m westward and the
second is 13.0 m northward. What are the
magnitude and direction of the third
displacement?