Vectors 3 - Bearsden Academy
Download
Report
Transcript Vectors 3 - Bearsden Academy
Higher Mathematics
Unit 3.1
Vectors
1.
Introduction
A vector is a quantity with both
magnitude and direction.
It can be represented
using a direct line
segment
AB
A
This vector is named
A B or u or u
u
B
2.
Vectors in 3 - Dimensions
3
v
5
2
........
2
v ........
5
........
3
P
Z
........
-2
O P ........
4
........
3
Y
Q
2
O
3
-3
3
-2
R
X
Z
Y
O
........
3
O Q ........
2
........
0
Q
X
-3
3
-2
R
Z
Y
O
X
........
3
O R ........
-3
........
R
-2
3.
Finding the components of a Vector
from Coordinates
y
P (1, 2)
Q (6, 3)
Q
3
2
P
1
6
x
..........
6 - 1 .......
5
PQ
3 - 2 .......
1
..........
y
S (-2, 1)
T (5, 3)
T
3
S
-2
1
5
x
..........
5 - -2 .......
7
PQ
3 - 1 .......
2
..........
y
A (-2, -1)
B
1
B (4, 1)
x
A
-2
-1
4
..........
4 - -2 .......
6
PQ
1- - 1 .......
2
..........
4.
Magnitude
4
4
AB u
-3
···
A
u
-3
B
AB u
2
2
4 + (-3)
···············
16 9
25 5
5.
Adding Vectors
7
BC
1
···
C
B
2
AB
4
···
1
CD
···
-6
A
D
B C C D
A D Add
A Bvectors
2 7 1 10
“ Nose-to-tail”
4 1 6 1
3
u
1
u+v
2
v
4
v
u
Add vectors
“ Nose-to-tail”
3
u v
1
5
5
2
4
4
AB u
3
A
4
B A u
3
A
-u
u
B
B
B A is the negative of A B
u is the negative of u
v
2
v
4
-2
v
-4
...
3
u
1
u
u + -v
u-v
-v
u v u v
Add the negative of the vector
“ Nose-to-tail”
3
1
1
3
2
4
The Zero Vector
2
v
4
2
v
4
v v
v v
-v
2 2
4 4
v
Back to the
start.
Gone nowhere
0
0
7.
Multiplication by a Scalar
1
v
2
v
2v
21
2v
2 2
2
4
2v has TWICE the MAGNITUDE of v,
but v and 2v have the SAME DIRECTION.
i.e. They are PARALLEL
8.
Position Vectors
y
4
p OP
2
....
P (4, 2)
p
O
x
The position vector of a point P is the vector from the origin O, to P.
The position vector O P is denoted by p
x
p
y
then the components of the position vector of P are
z
If P has coordinates (x , y , z)
9.
Collinear points
NOT collinear
A
E
D
AB
B
BC
C
Collinear
If
AB k BC
w h e re k 0
then the vectors are parallel and have
a point in common - namely B - ,
this makes them collinear
10.
Dividing lines in given ratios
“Section Formula”
Give up John, they are getting bored!!
11.
Unit Vectors
A unit vector is any vector whose
length (magnitude) is one
The vector
u
2
3
2
3
1
3
is a unit vector
since
2
u
u 1
2
2
2
1
3
3
3
2
There are three special unit vectors:
1
i 0
0
0
j 1
0
0
k 0
1
z
y
0, 0,1
k
j
i
0,1, 0
1, 0, 0
x
All vectors can be represented using
a sum of these unit vectors
P
Z
........
-2
O P ........
4
........
3
Y
O
X
OP
-2 i +4 j +3 k
12.
Scalar Product
The scalar product (or “dot” product) is a kind of
vector “multiplication”. It is quite different from any
kind of multiplication we’ve met before.
x1
x2
a
y
The scalar product of the vectors
1 and b y 2
z
z
1
2
is defined as:
a b a b co s
where is the angle
between the vectors,
pointing out from the
vertex
a
or
a b x1 x 2 y 1 y 2 z 1 z 2
b
Calculating the angle between two vectors
We have already seen that
Rearranging gives
a b a b co s
co s
ab
a b
And hence we can find the angle between two vectors
Some important results using the scalar product
1. The scalar product is a number not a vector
2. If either
3.
a 0
or
then a b 0
b 0
Perpendicular vectors:
Provided a and
b
are non zero then if
a b co s 0
then
co s 0
so
90
ie a and
4.
b
0
are perpendiculiar
a (b c ) a b a c