Physics is the Science of
Measurement
Length
Weight
Time
We begin with the measurement of length:
its magnitude and its direction.
Distance: A Scalar Quantity
 Distance is the length of the actual path
taken by an object.
distance = 20 m
A
B
A scalar quantity:
Contains magnitude
only and consists of a
number and a unit.
Displacement—A Vector Quantity
• Displacement is the straight-line
separation of two points in a specified
direction.
D = 12 m, 20o
A
q
B
A vector quantity:
Contains magnitude
AND direction, a
number, unit & angle.
(12 m, 300)
Distance and Displacement
• Displacement is the change of position
based on the starting point. Consider a
car that travels 4 m, E then 6 m, W.
D
Net displacement:
4 m,E
x = -2
x = +4
6 m,W
D = 2 m, W
What is the distance
traveled?
10 m !!
Identifying Direction
A common way of identifying direction
is by reference to East, North, West,
and South. (Locate points below.)
Length = 40 m
N
40 m, 50o N of E
W
60o
60o
50o
60o
E
40 m, 60o N of W
40 m, 60o W of S
S
40 m, 60o S of E
Identifying Direction
Write the angles shown below by using
references to east, south, west, north.
N
W
45o
E
50o
S
N
W
E
S
0 S of
50
Click
to Esee the Answers
450 W. of
. .N
Rectangular Coordinates
y
(-2, +3)
(+3, +2)
+
(-1, -3)
+
x
Reference is made to
x and y axes, with +
and - numbers to
indicate position in
space.
Right, up = (+,+)
-
Left, down = (-,-)
(+4, -3)
(x,y) = (?, ?)
Trigonometry Review
• Application of Trigonometry to Vectors
Trigonometry
R
y
q
x
y
sin q 
R
x
cos q 
R
y
tan q 
x
y = R sin q
x = R cos q
R2 = x2 + y2
Example 1: Find the height of a building
if it casts a shadow 90 m long and the
indicated angle is 30o.
The height h is opposite 300 and
the known adjacent side is 90 m.
opp
h
tan 30 

adj 90 m
0
h
300
h = (90 m) tan 30o
90 m
h = 57.7 m
Finding Components of Vectors
A component is the effect of a vector along
other directions. The x and y components of
the vector (R,q) are illustrated below.
x = R cos q
R
q
x
y
y = R sin q
Example 2: A person walks 400 m in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
N
R
q
x
400 m
y
30o
E
y=?
x=?
The x-component (E) is ADJ:
x = R cos q
The y-component (N) is OPP:
y = R sin q
E
Example 2 (Cont.): A 400-m walk in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
Note: x is the side
400 m
30o
y=?
x=?
E
x = (400 m) cos 30o
= +346 m, E
adjacent to angle 300
ADJ = HYP x Cos 300
x = R cos q
The x-component is:
Rx = +346 m
Example 2 (Cont.): A 400-m walk in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
Note: y is the side
400 m
30o
y=?
x=?
E
opposite to angle 300
OPP = HYP x Sin 300
y = R sin q
y = (400 m) sin 30o
The y-component is:
= + 200 m, N
Ry = +200 m
Example 2 (Cont.): A 400-m walk in a
direction of 30o N of E. How far is the
displacement east and how far north?
N
400 m
30o
Rx =
Ry =
+200 m
E
The x- and ycomponents are
each + in the
first quadrant
+346 m
Solution: The person is displaced 346 m east
and 200 m north of the original position.
Resultant of Perpendicular Vectors
Finding resultant of two perpendicular vectors is
like changing from rectangular to polar coord.
R
q
x
R x y
2
y
2
y
tan q 
x
R is always positive; q is from + x axis
Example 3: A woman walks 30 m, W;
then 40 m, N. Find her total displacement.
+40 m
40
0
tan  
;  = 59.1
30
R

q = 59.1o N of W
-30 m
R  (30)  (40)
2
2
R = 50 m
(R,q) = (50 m, 126.9o)
Component Method
1. Start at origin. Draw each vector to scale
with tip of 1st to tail of 2nd, tip of 2nd to
tail 3rd, and so on for others.
2. Draw resultant from origin to tip of last
vector, noting the quadrant of the resultant.
3. Write each vector in x,y components.
4. Add vectors algebraically to get resultant in
x,y components. Then convert to the total
vector (R,q).
Example 4. A boat moves 2.0 km east then
4.0 km north, then 3.0 km west, and finally
2.0 km south. Find resultant displacement.
1. Start at origin.
Draw each vector to
scale with tip of 1st
to tail of 2nd, tip of
2nd to tail 3rd, and
so on for others.
D
2 km, S
N
3 km, W
C
B
4 km, N
A
2 km, E
E
2. Draw resultant from origin to tip of last
vector, noting the quadrant of the resultant.
Note: The scale is approximate, but it is still
clear that the resultant is in the fourth quadrant.
Example 4 (Cont.) Find resultant displacement.
3. Write each vector
in i,j notation:
A = +2 x
B=
+4y
C = -3 x
D=
-2y
R = -1 x + 2 y
1 km, west and 2
km north of origin.
D
2 km, S
N
3 km, W
C
B
4 km, N
A
2 km, E
E
4. Add vectors A,B,C,D
algebraically to get resultant
in x,y components.
5. Convert to resultant
vector See next page.
Example 4 (Cont.) Find resultant displacement.
Resultant Sum is:
R = -1 x + 2 y
D
2 km, S
N
3 km, W
C
Now, We Find R, q
R  (1)  (2)  5
2
2
B
4 km, N
A
2 km, E
R = 2.24 km
2 km
tan  
1 km
 = 63.40 N of W
R

Rx = -1 km
Ry= +2
km
E
Conclusion of Chapter 3B - Vectors