Review
Average Acceleration Average Velocity
x
v
t
x  x f  x i
Displacement
v
a
t
Instantaneous velocity (acceleration) is the slope of the line tangent to the curve
of the position (velocity) -time graph

For constant acceleration…

For constant gravitational acceleration…
v  v 0  at
gt
vv vv00 gt
1 2
x  v 0 t  at
2
v 2  v 02  2ax
1 22
1
y vv00tt gt
gt
y
22
22  v22  2gy
v
v  v 00  2gy
Chapter 2
Motion in two dimensions
2.1: An introduction to vectors
Many quantities in physics, like displacement, have a
magnitude and a direction. Such quantities are called
VECTORS.
Other quantities which are vectors: velocity, acceleration,
force, momentum, ...
Many quantities in physics, like distance, have a
magnitude only. Such quantities are called SCALARS.
Other quantities which are scalars: speed, temperature,
mass, volume, ...
How can we find the magnitude if we have the
initial point and the terminal point?
The distance formula
Q
x2 , y2 
Terminal Point
Initial Point
x1, y1 
P
How can we find the direction? (Is this all looking familiar for each
application? You can make a right triangle and use trig to get the angle!)
Although it is possible to do this for any initial and
terminal points, since vectors are equal as long as
the direction and magnitude are the same, it is
easiest to find a vector with initial point at the
origin and terminal point (x, y).
Q
xx,
2 , yy
2
Terminal Point
A vector whose
initial point is the
origin is called a
position vector
Initial Point
0x1,, 0y1
P
If we subtract the initial point from the terminal
point, we will have an equivalent vector with initial
point at the origin.
Equality of Two Vectors
Two vectors are equal if they have
the same magnitude &
direction
Are the vectors here equal? 
A vector is a quantity that has both magnitude and direction. It is
represented by an arrow. The length of the vector represents the
magnitude and the arrow indicates the direction of the vector.
Blue and orange
vectors have same
magnitude but different
direction.
Blue and purple vectors
have same magnitude
and direction so they
are equal.
Blue and green vectors
have same direction but
different magnitude.
Two vectors are equal if they have the same direction and magnitude (length).
Addition of vectors
Given two vectors A & B , what is A  B
B



A
?
Graphical Techniques of Vector Addition
Two vectors can be added using these method:
1- tip to tail method.
2- the parallelogram method.
R
B

1-“Tip-to-Tail Method”
Two vectors can be added by •
placing the tail of the 2nd on
the tip of the 1st
A


B
R



A
Vector A
30 m
θ = 45O
To add the vectors
Place them head to tail
Vector B
50 m
θ= 0O
C
Vector C
30 m
Θ = 90O
B
A
Angle is measured at 40o
Resultant = 9 x 10 = 90 meters
ALL VECTORS MUST
BE DRAWN TO
SCALE & POINTED IN
THE PROPER
DIRECTION
A
D
R
B
C
B
C
A
D
A
+ B
+ C + D=
R
2- the parallelogram method.
2
2
C= A +B
Multiplying a Vector by a Scalar
• Given s , what is
?
3s  s  s  s

s

Scalar multiplication
Scalar multiplication: multiply vector by scalar
direction stays same
magnitude stretched by given scalar
(negative scalar reverses direction)


s
s
Vector Subtraction
A
-C
-D
B
B
R
A
=
C
D
=
-C
-D
A
A
+ B
+B + (
-
C
-
D=
-C ) + ( - D)
R
=
R
Example
Example
Example
Components of a Vector
Vector component:
A  Ax  Ay
where Ax and A y are the
components of the vector A




Unit Vectors
A unit vector is a vector that has a magnitude of 1, with no units.
Its only purpose is to point
We will use x , y for our Unit Vectors
x means x – direction, y is y – direction,
We also put little “hats” (^) on x , y to show that they are unit vectors
Notes about Components
•The previous equations are valid only if Ѳ is measured
with respect to the X-axis.
•The components can be positive or negative and will have the
same units as the original vector .
Vector component
at
VECTOR COMPONENTS
• WHAT ARE THE X AND Y COMPONENTS OF A VECTOR 40 m
θ=60O ?
• AX = 40 m x COS 600 = 20 m
• AY = 40 m x SIN 600 = 34.6 m
• WHAT ARE THE X AND Y COMPONENTS OF A VECTOR 60 m/s
θ = 2450 ?
• BX = 60 m/S x COS 245 0 = - 25.4 m/S
• BY = 60 m/S x SIN 245 0 = - 54.4 m/S
Example :
find the magnitude of the vector W
w  3i  4 j
w 
 3   4
2
What is w ?
2
 25  5
Example :
The angle between
Ax  25
&
A  Ax  Ay
Ay  45
and the positive x axis is:
1.
2.
3.
4.
5.
61°
29°
151°
209°
241°

where
Vector component:
If we want to add vectors that are in the form a i + b j, we can just add the
i components and then the j components.
v  2 i  5 j
Example :
w  3i  4 j
v  w   2i  5 j  3i  4 j  i  j
Let's look at this geometrically:
Can you see
from this picture
how to find the
length of v?
3i
w
5j
v
 2i i
 4j
j
When we want to know the
magnitude of the vector
(remember this is the length) we
denote it
v

 2  5
2
 29
2
ADDING & SUBTRACTING VECTORS USING COMPONENTS
ADD THE FOLLOWING
THREE VECTORS USING
COMPONENTS
Vector A
30 m
θ = 45O
Vector B
50 m
θ = 0O
Vector C
30 m
Θ = 9 0O
(1) RESOLVE EACH INTO
X AND Y COMPONENTS
ADDING & SUBTRACTING VECTORS USING
COMPONENTS
• AX = 30mx
cos 450 = 21.2 m
• AY = 30 m x sin 450 = 21.2 m
•BX = 50 m x cos 00 = 50 m
•BY = 50 m x sin 00 = 0 m
•CX = 30 m x cos 900 = 0 m
•CY = 30 m x sin 900 = 30 m
(2) ADD THE X COMPONENTS OF EACH VECTOR
ADD THE Y COMPONENTS OF EACH VECTOR
 X = SUM OF THE Xs = 21.2 + 50 + 0 = +71.2
 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2
(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE
 X AS THE BASE AND  Y AS THE OPPOSITE SIDE
 Y = +51.2
 X = +71.2
THE HYPOTENUSE IS THE RESULTANT VECTOR
(4) USE THE PYTHAGOREAN THEOREM TO THE LENGTH
(MAGNITUDE) OF THE RESULTANT VECTOR
angle
tan-1 (51.2/71.2)
Θ = 35.7 O
QUADRANT I
 Y = +51.2
 X = +71.2
(+71.2)2 + (+51.2)2 = 87.7
(5) FIND THE ANGLE (DIRECTION) USING INVERSE
TANGENT OF THE OPPOSITE SIDE OVER THE
ADJACENT SIDE
RESULTANT = 87.7 m
θ = 35.7 O
SUBTRACTING VECTORS USING COMPONENTS
Vector A
30 m
θ = 45O
A
A
-B +
+
(-
B
C
- Vector B
50 m
Vector C
30 m
θ = 90O
R
) +C =R
Vector A
30 m
θ = 45O
Vector B
50 m
θ = 0O
=
θ = 180O
Vector C
30 m
θ = 90O
•
RESOLVE EACH INTO X AND Y COMPONENTS
X-comp
y-comp
AX = 30 m x cos 450 = 21.2 m
•BX = 50 m x cos1800 = - 50 m
•CX = 30 m x cos 900 = 0 m
 X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8
 Y = +51.2
 X = -28.8
AY = 30 m x sin450 = 21.2 m
BY = 50 m x sin 1800 = 0
CY = 30 m x sin 900 = 30 m
 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2
(2) ADD THE X COMPONENTS OF EACH VECTOR
ADD THE Y COMPONENTS OF EACH VECTOR
 X = SUM OF THE Xs = 21.2 + (-50) + 0 = -28.8
 Y =SUM OF THE Ys = 21.2 + 0 + 30 = +51.2
(3) CONSTUCT A NEW RIGHT TRIANGLE USING THE
 X AS THE BASE AND  Y AS THE OPPOSITE SIDE
 Y = +51.2
 X = -28.8
THE HYPOTENUSE IS THE RESULTANT VECTOR
 Y = +51.2
 X = -28.8
R=
(-28.8)2 + (+51.2)2
58.7
RESULTANT ( R) = 58.7 m
angle
Θ=tan-1 (51.2/-28.8)
θ = -60.6 0
(1800 –60.60 ) = 119.40
QUADRANT II
=
θ = 119.4O
If we want to add vectors that are in the form a i + b j, we can just add the
i components and then the j components.
v  2 i  5 j
Example :
w  3i  4 j
v  w   2i  5 j  3i  4 j  i  j
Let's look at this geometrically:
Can you see
from this picture
how to find the
length of v?
3i
w
5j
v
 2i i
 4j
j
When we want to know the
magnitude of the vector
(remember this is the length) we
denote it
v

 2  5
2
 29
2
example
Example :
If we know the magnitude and direction of the vector, let's
see if we can express the vector in a + b form.
Example :
v  5,   150
As usual we can use the trig we know
to find the length in the horizontal
direction and in the vertical direction.
5
150
v  5cos 150 xˆ  sin 150 yˆ   
5 3
5
ˆx  yˆ
2
2
Example :
F1 = 37N 54° N of E
F=F1+F2+F3
F2 = 50N 18° N of W
F3 = 67 N 4° W of S
Ex : 2 – 10
A woman walks 10 Km north, turns toward the north west , and
walks 5 Km further . What is her final position?
Example :
Example :
Example :