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

This vector is named

or or ��������������AB u uA

B

u

2.Vectors in 3 - Dimensions

v

2

5

3

........

........

........

v2

5

3

3-3

2

3-2

P

Q

R

O X

Y

Z........

........

........

OP

�������������� -2

4

3

-33

-2

Q

R

O

Z

X

Y

........

........

........

��������������OQ

3

2

0

Z

X

R

O

Y

........

........

........

��������������OR

3

-3

-2

3. Finding the components of a Vector

from Coordinates

x

y

P

Q

.......... .......

.......... .......PQ

��������������

P (1, 2)

Q (6, 3)

6 - 1

3 - 2

5

1

1 6

32

x

y

S

T

.......... .......

.......... .......PQ

��������������

S (-2, 1)

T (5, 3)

5 - -2

3 - 1

7

2

5 -2

3

1

x

y

A

B

.......... .......

.......... .......PQ

��������������

A (-2, -1)

B (4, 1)

4 - -2

1- - 1

6

2

-2 4

- 1

1

4.Magnitude

···AB u

��������������u

A

B

AB u��������������

16 9

25 5

··············· 42 (-3)2+

4

-34-3

5.Adding Vectors

A

B

C

D

2

7

···CD

�������������� 1-6

AD ��������������

BA��������������

CB��������������

DC��������������

1

6

2

4

7

1

10

1

···AB

��������������

···BC

��������������

1

4

Add vectors

“ Nose-to-tail”

u

u + v

3

1u

Add vectors

“ Nose-to-tail”

v

2

4v

3 2

1 4u v

5

5

4

3AB u

��������������

u

A

B

-u

A

B

4

3BA u

��������������

BA��������������

is the negative of AB��������������

u is the negative of u

u

3

1u

Add the negative of the vector

“ Nose-to-tail”

-v

2

4v

u v u v

1

3

v

u + -vu - v

...v

-2

-4

23

1 4

v

The Zero Vector

v v

-v

2

4v

2

4v

v v 22

4 4

0

0

Back to the

start.Gone nowhere

7.Multiplication by a Scalar

1

2v

v2v

22

2

1

2v

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

x

y

O

P (4, 2)

x

y

O

p

P (4, 2)

The position vector of a point P is the vector from the origin O, to P.

OP��������������

pThe position vector is denoted by

....p OP

�������������� 42

x

p y

z

If P has coordinates (x , y , z)

then the components of the position vector of P are

9.Collinear points

x

y

O

P (4, 2)

A

B

ED

C

AB��������������

BC��������������

NOT collinear

Collinear If where 0AB kBC k ����������������������������

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”

x

y

O

P (4, 2)

Give up John, they are getting bored!!

11.Unit Vectors

A unit vector is any vector whose length (magnitude) is one

The vector

2

32

31

3

u is a unit vector since

2 2 22 2 1

3 3 3

u

1u

There are three special unit vectors:

1

0

0

i

0

1

0

j

0

0

1

k

x

z

y

i 1,0,0

j 0,1,0k

0,0,1

All vectors can be represented using a sum of these unit vectors

........

........

........

OP

��������������

OP ��������������

P

O X

Y

Z-2

-2i

4

j+4

3

+3k

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.

The scalar product of the vectors and

is defined as:

1

1

1

x

a y

z

2

2

2

x

b y

z

cosa b a b

1 2 1 2 1 2a b x x y y z z or

where is the angle between the vectors, pointing out from the vertex

a

b

Calculating the angle between two vectors

We have already seen that cosa b a b

Rearranging givescos

a b

a b

And hence we can find the angle between two vectors

Some important results using the scalar product

3. Perpendicular vectors:

Provided and are non zero then if

then

so

ie and are perpendiculiar

a b

1. The scalar product is a number not a vector

0a b 0b0a 2. If either or then

cos 0a b

cos 0 090

a b

4.

( )a b c a b a c