50: Vectors50: Vectors

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

Vectors

A vector has magnitude (Size) and direction.

Examples of vectors are weight, force, velocity and momentum.Drawing Vectors

Magnitude is shown by the length of a line.Direction is shown by an arrow.

This vector . . .

A

BABis written

as

Vectors

Vectors are parallel if one is a multiple of the other.

A

B

A

B

BA

AB

PQCD 2

P

Q

C

D

VectorsPosition VectorsA position vector gives the position of a point relative to the origin, O.

e.g.

A position vector is usually written with a single letter.

)4,3(A

O x

x

3

4

4

3OA

Vectors

)4,3(A

O x

x

4

3OA a

Position VectorsA position vector gives the position of a point relative to the origin, O.

e.g.

An arrow is not used but the letter must be

underlined.In typed work, the single letter used for a

position vector is typed in bold and not underlined.

3

4

4

3 is called the column vector or component form.

Vectors

x

y

z

O

Another NotationVectors can be given in terms of unit vectors.A unit vector has magnitude 1 and those we use are in the directions of the axes.

i

k

1

2

3e.g. kji 23

If you don’t like writing i, and k all the

time, just switch to column vectors.

j

j

In 3 dimensions the unit vectors are labelled i, and k along the x-, y- and z- axes respectively.

j

Vectors

ab

a

A

Ob

B

b a

Think of this as walking along the vector. To get from A to B we could go

from A to O: then from O to B:

If A and B have position vectors a and b respectively,

can be written in terms of a and b.AB

AB

aOAAO

bOB

- a

AB

b aSo

Vectors

ab

a

A

Ob

B

b a

If A and B have position vectors a and b respectively,

can be written in terms of a and b.AB

AB- a

AB

So

Rule AB b`s - a`suuur

ab

PQ q`s - p`s = q - puur

Vectors

b - aAB

e.g. If A and B are given by

express as a column vector. AB

)2,1()4,3( BA and

,4

3

aSolution

:

2

1b

4

3

2

1

6

4

a

A

Ob

B

-a + b

Vectors

)3,1(A

)1,4( B

The magnitude is given by the length AB.

Magnitude of a Vectore.g.

ab

3

1

1

4

The vector AB

4

3

so, using Pythagoras’ theorem,

3

4

222 43 AB5 AB

The arrow shows the direction.The length is 4.

As we are squaring each component, we can ignore any minus signs.

Vectors

To find the magnitude of a 3 dimensional vector, we extend Pythagoras’ theorem.

Magnitude of a Vector

AB

3

2

4If

4A

B

2C 222 24 AC

e.g.

Vectors

AB

3

2

4If

222 24 AC

Magnitude of a Vector

3

222 CBACAB 2222 3)24( AB

29 AB

C

A

B

To find the magnitude of a 3 dimensional vector, we extend Pythagoras’ theorem.e.g.

Vectors

Notation

The magnitude of a vector can be written in a number of ways:

The magnitude of is written asAB

orAB AB

The magnitude of is written as or a a a

VectorsMagnitude of a Vector

AB

b

a

222 cba

22 ba

In general, if

AB

c

b

a

the magnitude AB =

the magnitude AB =

As we are squaring each component, we can ignore any minus signs.

VectorsThe mid-point of a

vector

a

A

Ob

BMx

If M is the mid-point of AB,

)(21 aba

aba 21

21

ABOA 21

mOMThe position vector of the mid-point of is given by the average of the position

vectors of A and B.

AB

)(21 ba m

Vectors

cd CD

e.g. If C and D are given by

express as a column vector. CD

)2,0,1()2,2,1( DC and

,

2

2

1

c

Solution:

2

0

1

d

2

2

1

2

0

1

A huge advantage of vectors over trig or coordinate geometry is that working in 3 dimensions is almost as easy as 2.

0

2

2

Vectors

ab AB

SUMMARY

• A position vector gives the position of a point relative to the origin, O.

a

O x

Ax

• The vector is given by

AB

b

Bx

• The mid-point of is given byAB

)(21 ba

( the average )

VectorsExercise1. If the points A, B and C are given by

find (a) (b) (c)

)2,3(B )3,4(Cand

AB

,

3

4

1

p

AB

BC

AC

(d) the magnitude of

2. Find the vector and the magnitude of if the position vectors of P and Q are given by

PQ

1

1

2

q

PQ

,)1,1( A

VectorsSolutions

,)1,1( A1.

)2,3(B )3,4(Cand

(a) abAB

1

1

2

3

3

4

2

3

3

4

1

7

1

1

3

4

4

3

BC bc (b)

AB(d) the magnitude

of 534 22

ac (c) AC

Vectors

,

3

4

1

p

2. Find the vector and the magnitude of if the position vectors of P and Q are given by

PQ

1

1

2

q

PQ

Solution:

PQ

pq

3

4

1

1

1

2

2

5

3

222 253 PQ 38

Vectors

AB

3

4

1

7BC

BCABAC

In the previous exercise we had

4

3AC

and

So,

This result can be extended to any number of vectors.

Adding Vectors

Vectors

e.g.

A B

C

D

CDBCABAD

Can you see what equals? DACDBCAB

ANS: 0 DACDBCAB

VectorsExercise

ia 3

kjif 23

kib 3

1. The diagram shows a cuboid. The position vectors of A, B and F relative to O are given by

x

y

z

O A

BC

G F

ED

(a) Find the position vectors of C, D, E and G and the mid-point of BD.

(b) Find the vectors , and .

FG

AG

AF

VectorsSolution:

ia 3

kjif 23

kib 3

x

y

z

O A

BC

G FED

jkCGOCg 2

jiAEOAe 23

jbfBFODd 2

kabABOCc ( since OC

AB )

(a) Find the position vectors of C, D, E and G and the mid-point of BD.

Mid-point of BD

)23()( 21

21 kjidb

You could get this answer, and the following

ones, by just looking at the diagram. OC is

parallel to the

z-axis so equals the length OC multiplied by k.

31

2

VectorsSolution:

ia 3

kjif 23

kib 3

x

y

z

O A

BC

G FED

31

2

OGAF

kj 2

(b) Find the vectors , and .

FG

AG

AF

Vectors

FG

Solution:

ia 3

kjif 23

kib 3

x

y

z

O A

BC

G FED

31

2

OGAF

kj 2

i3AOFG

(b) Find the vectors , and .

AG

AF

VectorsSolution:

ia 3

kjif 23

kib 3

x

y

z

O A

BC

G FED

31

2

OGAF

kj 2

i3AOFG

i3OGAOAG

kj 2

(b) Find the vectors , and .

FG

AG

AF

Vectors

Other Unit VectorsWe can easily find a unit vector in the direction of a given vector by dividing by the magnitude.

403

a

e.g. Find a unit vector in the direction of a where

Solution:

543 22 aa

The unit vector,

403

5

1a

Vectors

Vectors

The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.

Vectors

ab AB

SUMMARY

• A position vector gives the position of a point relative to the origin, O.

a

O x

Ax

• The vector is given by

AB

b

Bx

• The mid-point of is given byAB

)(21 ba

( the average )

Vectors

Adding Vectors

CDBCABAD

0 DACDBCAB

e.g.

and

A B

C

D

Vectors

Notation

The magnitude of a vector can be written in a number of ways:

The magnitude of is written asAB

orAB AB

The magnitude of is written as or a a a

VectorsMagnitude of a Vector

AB

b

a

222 cba

22 ba

In general, if

AB

c

b

a

the magnitude AB =

the magnitude AB =

As we are squaring each component, we can ignore any minus signs.

Vectors

kx

y

z

O

Another NotationVectors can be given in terms of unit vectors.A unit vector has magnitude 1 and those we use are in the directions of the axes.

1

2

3e.g. kji 23

If you don’t like writing i, and k all the

time, just switch to column vectors.

j

In 3 dimensions the unit vectors are labelled i, and k along the x-, y- and z- axes respectively.

j

k

ij

Vectors

Other Unit VectorsWe can easily find a unit vector in the direction of a given vector by dividing by the magnitude.

403

a

e.g. Find a unit vector in the direction of a where

Solution:

543 22 aa

The unit vector,

403

5

1a