vectors intro
TRANSCRIPT
Block 3
Vectors Introduction
What is to be learned?
• What a vector is• Some other bits and pieces about vectors
A vector is
Like a journey
Need DistanceDirection
- magnitude
A
B
A
4
2
42( )B BA
4–
2–
-4 -2( ) components
A
B
– 3
AB
4
-3 4
( )
u˜
u˜ = =
C
D
-2
DC
4
4 -2
( )
t
t˜
= =
3 4( )u =
˜U = (3 , 4)
x
y U
u˜
u˜
u˜
components and coordinates
A
B
– 4
AB
2
-4 2
( )
u˜
u = =
Magnitude of a Vector
5
-4
5 -4( )
v˜
v =˜
magnitude of v =˜
√(52 +(-4)2))
5
-4
5 -4( )
v˜
v =˜
v =˜
√(52 +(-4)2))| |
3
-6
-6 3
( )
t
t˜ =
=t ?| | t˜| |
√((-6)2 + 32)
= √45
Vectors
• Describe a “journey”– Distance– Direction
• Numbers work like coordinates– Called components
• The magnitude of vector is calculated using pythagoras
magnitude
a b
( )u =˜
| u2 | =˜
a2 + b2
A
B
4
-3
4 -3( )AB =
u˜
= u˜
| u2 | =˜
42 + (-3)2
| u |˜
= 5
C
D
2
DC
-5
-5 2
( )
t
t˜
= = =t˜| |
√((-5)2 + 22)
= √29
Name the vector in two ways, write itscomponents and find its magnitude.
Key Question.
What is to be learned?
• How to add and “subtract” vectors.• How to multiply a vector by a scalar.
Vector Addition
a + b42( )a = b =
a+b
nose to tail
1 3
( )
a
b
5 5
= ( )
a + b ?( )a = b =( )
= ( )
4 5
2 3
( )4 5
( )2 3+
6 8
4 + 2
5 + 3
using components
laughably easy
Vector Subtraction
vector subtraction does not exist!(or does it?)
9 – 7= 9 + (-7)
we add the negative of the vector
5
–
5 -4( )
v˜
v =˜
- v =˜
–
–
4
-5 4
( )
a – b ?( )a = b =( )
= ( )
4 6
2 -3
-b =( )-2 3
a + -b
( )4 6
( )-2 3
+
2 9
4 – 2
6 – (-3)
Multiplying by a scalar
2 -1( )a = 3a
a
= a + a + a
a
a
3a = 6 -3( )
3a
Addition, Subtraction and Multiplication (by a Scalar)
Normal Number
a + b52( )a = b =
a+b
nose to tail
1 3
( )
a
b
6 5
= ( )adding
a – b ?( )a = b =( )= ( )
7 6
4 1
3 5
subtracting Theory – Add the negativePractice -
Scalar Multiplication
( )a = 1 2
3a?
3a = ( )3 6
a + b-5 2( )a = b =
a+b
2 3
( )
a
b
-3 5 = ( )
Calculate a + b, then represent the additiondiagrammatically.Key Question.