vectors (1) units vectors units vectors magnitude of vectors magnitude of vectors
TRANSCRIPT
Vectors (1)Vectors (1)•Units VectorsUnits Vectors
•Magnitude of VectorsMagnitude of Vectors
A
B
The vector AB
The vector a
a
Notation
( )85
… as a column vector
8 across, 5 up
The vector a
a
The vector 2 a
a
Notation
Is twice as long as a,but in the same direction
2a a
a
This bit is a scaler
Displacement“a measure of distance and direction”
60o
50m
NAn object moves 50m at 60o to the East-West x-axisHow far East has it gone?
How far North has it gone?
N
E
Cos 60o = adj/hyp = E/50E = 50 cos 60o = 25m
Sin 60o = opp/hyp = N/50N = 50 sin 60o = 43.3m (1 d.p.)
This can be expressed as a column vector:-
Displacement = [ ]2543.3
Unit Vectors (1)
-1
0
1
2
3
-1 0 1 2 3
X
Y
ii is the unit vector in
the x-direction
jj is the unit vector in
the y-direction
i = [ ]10
j = [ ]01
All vectors can be expressed as a linear combination of these 2 vectors
[ ] = 25 + 43.3e.g. displacement = 2543.3 [ ]1
0 [ ]01
Unit Vectors (2)
i = [ ]10 j = [ ]0
1
All vectors can be expressed as a linear combination of these 2 vectors
[ ] = 25 + 43.3e.g. displacement = 2543.3 [ ]1
0 [ ]01
= 25 i + 43.3 jThis is the standard way displacement vectors are presented
Magnitude of a vectorThe displacement of a boat is given
by :- -10 i + 15 j
What is it’s magnitude ?
10
15
-10 i + 15 j By Pythagoras, the magnitude = (152 + 102) = 325 = 18.0 (1 d.p.)
The displacement is 18.0m
Magnitude
a = -10 i + 15 j
a
-10 i + 15 j By Pythagoras, the magnitude = (152 + 102) = 325 = 18.0 (1 d.p.)
a is notation for magnitude
a = 18.0
10
3
4
a
4
3
10a
Magnitude of a 3D Vector (1)
x
y
z
o
x
y
z
10
3
4
a
4
3
10a
Magnitude of a 3D Vector (2)
o B
3
By Pythagoras, OB = (32 + 42)
x
y
z
10
3
4
a
4
3
10a
Magnitude of a 3D Vector (3)
o B
3
By Pythagoras, OB = (32 + 42) A
(32 + 42)
By Pythagoras, OA2 = AB2 + OB2
AB2 = 102
OB2 = (32 + 42) OA2 = 102 + 32 + 42
OA = (102 + 32 + 42)
|a| = OA= 11.2
x
y
z
t
s
r
a
r
s
ta
Magnitude of a 3D Vector (General)
o|a| = (r2 + s2 + t2)
“the magnitude is the square root of the sum of the squares of the 3 components.”
[Pythagoras in 3D]