Chapter 3: Two –
Dimensional Motion
and Vectors
Section 3-1 and 3-2
pages 84-97
Vectors
 A scalar is a quantity that does not involve
direction.


55 mph
18 cm long
 A vector is a quantity that involves both
magnitude and direction (velocity, acceleration,
displacement, force)


55 mph north
A downward force of 3 Newtons
Definition
Magnitude R is represented
by length
Head
Direction θ is represented
by the angle
Tail
θ
The resultant vector can be defined in polar coordinates
as R at θ N of E.
Try YOURS!!
Parallel Vector Addition
Adding vectors in the same direction
10
23 (resultant)
=
13
Adding vectors in the opposite direction
5
= 3 (resultant)
8
Basic Trig Functions
x
R= hyp
θ
90o
A = adj
For the right triangle placed at the origin
Sin θ = B/R = opp/hyp
Cos θ = A/R = adj/hyp
Tan θ = B/A = opp/adj
A2 + B2 = R2
B = opp
y
Perpendicular Vector Addition
For two perpendicular vectors
5
12
Construct resultant R by drawing a vector from the tail of the horizontal
vector to head of the vertical vector
R
R2 = 52 + 122
R = 13
θ
12
5
θ = tan-1 5/12 = 22.6o
Example:
A boat heads east at 8.00 m/s across a river flowing
north at 5.00 m/s. What is the resultant velocity of
the boat?
5.00 m/s N
Ө
8.00 m/s E
1) Use pythagorean theory.
2) Use tan Ө = opp/hyp
R = 9.43 m/s at 32°
Multiple Vector Addition
A
R
E
B
C
D
R
C
E
B A
D
Can be added in any order!!
A + B + C + D + E = Distance
R = Resultant = Displacement
Adding Vectors SUMMARY
The sum of two or more vectors is known as the
RESULTANT
Vectors Acting in the Same Direction
ADD
(parallel)
Vectors Acting in the Opposite Direction (parallel)
SUBTRACT
At 90o angles – Ah- Trigonometry. . .
PYTHAGOREAN
TAN Ө
(perpendicular)
At angles other than 90o - three methods
1. Graphical – scaled drawing
2. Resolution into Components Method –
break each vector into right triangles then
use trig functions
3. Law of Sines and Cosines
c2 = a2 + b2 – 2abcosC
a
=
b
sin A
sin B
=
c
sin C
Using the Graphical Method of
Vector Addition:
 Vectors are drawn to scale and the resultant is
determined using a ruler and protractor.
 Vectors are added by drawing the tail of the second
vector at the head of the first (tip to tail method).

The order of addition does not matter.
 The resultant is always drawn from the tail of the
first to the head of the last vector.
BE METICULOUS IN YOUR DRAWING!!! Your
accuracy depends on it. (±2°, 0.2 cm)
Method 1: Adding Vectors Graphically
(It’s making a scaled drawing.)
Steps:
Decide what quadrant the vectors will be in. Draw the axis and write the SCALE in a box.
Draw the first vector to scale starting at the origin and label it
Draw the remaining vectors, so that they make a
B
C
D
_____,
______,
_____,
etc.
Draw the
it R
A
.
TAIL TO HEAD
RESULTANT as the dashed line from the
path and label them
TAIL OF 1ST VECTOR TO
HEAD OF LAST
and label
.
R to get the
Measure the length of ____
MAGNITUDE
(from the closest axis) to get the
DIRECTION
and the angle of R
and write your answer in a box.
DIRECTION ALWAYS < 45° angle
of
.
Example:
Cartman gets upset with Kenny for taking his doughnut. Cartman
chases Kenny 30 meters at 40o N of E and then 20 meters at 10o
E of N. Calculate Cartmans’s total displacement. Solve this
graphically.
Example:
Cartman gets upset with Kenny for taking his doughnut. Cartman
chases Kenny 30 meters at 40o N of E and then 20 meters at 10o
E of N. Calculate Cartmans’s total displacement. Solve this
graphically.
Advantages and Disadvantages
of the Graphical Method
 Can add any
 Must be correctly
number of vectors
at once
 Uses simple tools
 No mathematical
equations needed
draw to scale and
at appropriate
angles
 Subject to human
error
 Time consuming
This completes Method One!
 So lets get
 Vector problems #1 and #2 due
tomorrow.