Vectors (1)
•Units
Vectors
•Magnitude of Vectors
Notation
B
The vector AB
A
… as a column vector
()
8
5
8 across, 5 up
a
The vector a
Notation
a
The vector a
2a
a
This bit is
a scaler
a
The vector 2 a
Is twice as long as a,
but in the same direction
Displacement
“a measure of distance and direction”
An object moves 50m at 60o
to the East-West x-axis
N
How far East has it gone?
50m
Cos 60o = adj/hyp = E/50
E = 50 cos 60o = 25m
N
60o
How far North has it gone?
E
Sin 60o = opp/hyp = N/50
N = 50 sin 60o = 43.3m (1 d.p.)
This can be expressed as a column vector:-
Displacement =
[]
25
43.3
Unit Vectors (1)
i is the unit vector in
3
i
the x-direction
2
j
1
Y
i=
j is the unit vector in
the y-direction
0
-1
[]
1
0
0
1
2
3
j=
-1
X
[]
0
1
All vectors can be expressed as a
linear combination of these 2 vectors
e.g. displacement =
[]
25
43.3
= 25
[ ] + 43.3 [ ]
1
0
0
1
Unit Vectors (2)
i=
[]
1
0
j=
[]
0
1
All vectors can be expressed as a
linear combination of these 2 vectors
e.g. displacement =
[]
25
43.3
= 25
[ ] + 43.3 [ ]
1
0
= 25 i + 43.3 j
This is the standard way displacement
vectors are presented
0
1
Magnitude of a vector
The displacement of a boat is given by :-
-10 i + 15 j
What is it’s magnitude ?
-10 i + 15 j
15
By Pythagoras,
the magnitude = (152 + 102)
= 325 = 18.0 (1 d.p.)
The displacement is 18.0m
10
Magnitude
a = -10 i + 15 j
a
is notation for magnitude
-10 i + 15 j
a
By Pythagoras,
the magnitude = (152 + 102)
= 325 = 18.0 (1 d.p.)
a
= 18.0
Magnitude of a 3D Vector (1)
z
a
10
3
o
4
x
y
4
 
a 3 
10 
 
Magnitude of a 3D Vector (2)
z
a
10
3
o
B
4
3
x
y
4
 
a 3 
10 
 
By Pythagoras,
OB = (32 + 42)
Magnitude of a 3D Vector (3)
z
By Pythagoras,
OB = (32 + 42)
A
4
 
a 3 
10 
 
a
10
3
o
(32 + 42)
4
B
3
x
y
By Pythagoras,
OA2 = AB2 + OB2
AB2 = 102
OB2 = (32 + 42)
OA2 = 102 + 32 + 42
OA = (102 + 32 + 42)
|a| = OA = 11.2
Magnitude of a 3D Vector (General)
r
 
a  s
t 
 
z
a
t
y
|a| = (r2 + s2 + t2)
s
o
“the magnitude is the square root
of the sum of the squares of
the 3 components.”
r
x
[Pythagoras in 3D]