vectors intro
TRANSCRIPT
![Page 1: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/1.jpg)
Block 3
Vectors Introduction
![Page 2: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/2.jpg)
What is to be learned?
• What a vector is• Some other bits and pieces about vectors
![Page 3: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/3.jpg)
A vector is
Like a journey
Need DistanceDirection
- magnitude
![Page 4: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/4.jpg)
A
B
A
4
2
42( )B BA
4–
2–
-4 -2( ) components
![Page 5: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/5.jpg)
A
B
– 3
AB
4
-3 4
( )
u˜
u˜ = =
![Page 6: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/6.jpg)
C
D
-2
DC
4
4 -2
( )
t
t˜
= =
![Page 7: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/7.jpg)
3 4( )u =
˜U = (3 , 4)
x
y U
u˜
u˜
u˜
components and coordinates
![Page 8: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/8.jpg)
A
B
– 4
AB
2
-4 2
( )
u˜
u = =
![Page 9: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/9.jpg)
Magnitude of a Vector
![Page 10: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/10.jpg)
5
-4
5 -4( )
v˜
v =˜
magnitude of v =˜
√(52 +(-4)2))
![Page 11: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/11.jpg)
5
-4
5 -4( )
v˜
v =˜
v =˜
√(52 +(-4)2))| |
![Page 12: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/12.jpg)
3
-6
-6 3
( )
t
t˜ =
=t ?| | t˜| |
√((-6)2 + 32)
= √45
![Page 13: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/13.jpg)
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
![Page 14: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/14.jpg)
A
B
4
-3
4 -3( )AB =
u˜
= u˜
| u2 | =˜
42 + (-3)2
| u |˜
= 5
![Page 15: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/15.jpg)
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.
![Page 16: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/16.jpg)
What is to be learned?
• How to add and “subtract” vectors.• How to multiply a vector by a scalar.
![Page 17: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/17.jpg)
Vector Addition
![Page 18: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/18.jpg)
a + b42( )a = b =
a+b
nose to tail
1 3
( )
a
b
5 5
= ( )
![Page 19: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/19.jpg)
a + b ?( )a = b =( )
= ( )
4 5
2 3
( )4 5
( )2 3+
6 8
4 + 2
5 + 3
using components
laughably easy
![Page 20: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/20.jpg)
Vector Subtraction
![Page 21: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/21.jpg)
vector subtraction does not exist!(or does it?)
9 – 7= 9 + (-7)
we add the negative of the vector
![Page 22: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/22.jpg)
5
–
5 -4( )
v˜
v =˜
- v =˜
–
–
4
-5 4
( )
![Page 23: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/23.jpg)
a – b ?( )a = b =( )
= ( )
4 6
2 -3
-b =( )-2 3
a + -b
( )4 6
( )-2 3
+
2 9
4 – 2
6 – (-3)
![Page 24: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/24.jpg)
Multiplying by a scalar
![Page 25: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/25.jpg)
2 -1( )a = 3a
a
= a + a + a
a
a
3a = 6 -3( )
3a
![Page 26: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/26.jpg)
Addition, Subtraction and Multiplication (by a Scalar)
Normal Number
![Page 27: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/27.jpg)
a + b52( )a = b =
a+b
nose to tail
1 3
( )
a
b
6 5
= ( )adding
![Page 28: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/28.jpg)
a – b ?( )a = b =( )= ( )
7 6
4 1
3 5
subtracting Theory – Add the negativePractice -
![Page 29: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/29.jpg)
Scalar Multiplication
( )a = 1 2
3a?
3a = ( )3 6
![Page 30: Vectors intro](https://reader031.vdocuments.us/reader031/viewer/2022022202/587978321a28ab37368b7ab1/html5/thumbnails/30.jpg)
a + b-5 2( )a = b =
a+b
2 3
( )
a
b
-3 5 = ( )
Calculate a + b, then represent the additiondiagrammatically.Key Question.