Vectors

Readings: Chapter 3

Vectors

Vectors are the objects which are characterized by two parameters:

• magnitude (length)

• direction

These vectors are equal

a

a

a

Vectors can be considered as a set of two points in some space:

Point A is an initial point (starting position) and Point B is a final point (ending position)

a

A

B

Vectors: example

Vector: displacement of a car: initial point A, final point B

AB

Vector: velocity of a car, direction is the direction of the motion

The speed of the car could be the same (the magnitude of velocity could be the same) but the directions of the motion will be different - different vectors -velocity.

AB

Vectors

Vectors are characterized by two parameters:

• length (magnitude)

• direction

These vectors are the same

a

a

a

Sum of two vectors

Sum of two vectors:

b a

a b

a b

a

b a

b

Sum of two vectors: net displacement

B

A

C

Sum of vectors

Sum of vectors: for many vectors the procedure is straightforward

a b

a b c a

b

c

c

Vectors: multiplication by number

2a a a a

a

Vector (where is the

positive number) has the

same direction as , but

its magnitude is times

larger

c a

c

a a

2a

c

Vector (where is the

negative number) has the

direction opposite to ,

and times larger length

c a

c

a

c

a

2a

( ) 0a a a a

a

a

Vectors: multiplication by number

a

Vectors: Examples

The magnitude of is 5

What is the direction and the magnitude of

a

0.2b a

The magnitude of is , the direction is opposite to ab

0.2 5 1b

b

Vectors: coordinate system and vector component

y

x

A

coordinate system:

O (origin) 0xA

0yA

B

0xB

0yB

Position of point A is characterized by 2 numbers ,x yA A

Vectors: coordinate system and vector components

y

x

A

coordinate system:

O (origin) 0xA

0yA

B

0xB

0yB

,x yA Aa

Vector is characterized by two numbers

a

The meaning of this is the following:

x y x ya A A A i A j

xA

a

yA

i

j

,i j

- vectors with unit magnitude

Vectors: coordinate system and vector components

Then the sum of the vectors is the sum of their components:

1 1 2 2( , )a b a b a b 1 2( , )a a a

1 2( , )b b b

1 2( , )a a a

1 2( , )ca ca ca

1a

a

2a

i

j

1 2 1 2 1 1 2 2( ) ( )a b a i a j b i b j a b i a b j

Magnitude of the vector: 2 21 2a a a

Direction of the vector: 1a

a

2a

i

j

2

1

tana

a

Vectors: Example

Find the magnitude and direction of the sum of three vectors with components

(5,2) (-1,4) and (0,-2)

1 5 1 0 4d

Magnitude 2 2 2 21 2 4 4 5.6d d d

Direction1d

d

2d

i

j

2

1

tan 1d

d

d a b c

Components of vector d

2 2 4 2 4d

045

a

b

c

d