teach a level maths vectors for mechanics. volume 4: mechanics 1 vectors for mechanics

Teach A Level MathsTeach A Level Maths

Vectors for Vectors for MechanicsMechanics

Volume 4: Mechanics 1Volume 4: Mechanics 1Vectors for MechanicsVectors for Mechanics

Many of the quantities in Mechanics are vectors.This presentation covers the vector theory that you need in M1.

The theory is in sections so that, if you wish, you can do the parts as you need them.

1. Introducing Vectors

3. Position Vectors and Displacement

2. The Unit Vectors andi j

4. Column Vectors

Many of the quantities in Mechanics are vectors.This presentation covers the vector theory that you need in M1.

The theory is in sections so that, if you wish, you can do the parts as you need them.

1. Introducing Vectors

3. Position Vectors and Displacement

2. The Unit Vectors andi j

4. Column Vectors

Click on the section you want.

1. Introducing Vectors

A vector can be shown by a line segment with an arrow.

This vector is written as

ABA

B

A

B

BA AB

The arrow for runs from B to A.

BA A

B

BA

AB

O

The magnitude ( size ) of a vector is shown by the length of the line.

The grid has 1 cm squares.

P Q

AB has magnitude 3.We write |AB| 3.

B

The magnitude of . . .

PQ

is given by |PQ| 4.

A

R

O

S

We can use Pythagoras’ theorem to find the magnitude of other vectors.

Using Pythagoras’ theorem, RS

2 13

RS 3·61 ( 3 s.f. )3

2

32 + 22

e.g. AB

A

B

is equivalent to the sum of any vectors starting at A and ending at B.

AB P

PBAP Notice the directions of the arrows on the vectors.When we draw a vector and when we write it,

the “head” always points towards the 2nd letter.

head oftail of

APPB

and are drawn head-to-tail.

They are added to give .

AP PBAB

EXERCISE

(i)

B

(ii)

(v)

A

(iv) (vi)

B

C

(iii)

Answers:1(a) (v) (b) (ii)

(c) (iv)

2. What is the magnitude of (a) (b)

(c) AB BCAB BC

1. Which vector in the diagram is equal to

(a)(b) BA BA (c) AB BC (d) BC

(d) (i)2(a) 4 (b) 3 (c) 5

The next section looks at Unit Vectors.

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Continue to Unit Vectors

2. The Unit Vectors andi j

j

i

Instead of drawing diagrams to show vectors we can use unit vectors. They have magnitude 1.

e.g. A velocity v is given by v3 4 i j

x

y

j

i3

4v

The unit vectors and are parallel to the x- and y-axes respectively.

i j

In text-books single letters for vectors are printed in bold but we must underline them.

v3 4 i j

x

y

3

4v

|v|32 42

j

i

No or in magnitude

i j

|v|32 42

So, if we have the unit vector form, we use the numbers in front of and

i j

|v|5

Tip: Squares of real numbers are always positive so we never need any minus signs.

We can write |v| for speed.

The magnitude of velocity is speed, so, using Pythagoras’ theorem,

x

y

3

4v

j

i

v i j

The direction of the vector is found by using trig.

tan

53·1 ( 3 s.f. )

BUT beware !

3 443

If we need the direction of a vector when unit vectors are used, we must sketch the vector to show the angle we have found.

v3 4 i jSupposeWithout a diagram we get

tan

34

53·1 ( 3 s.f. )

So again

But, the vectors are not the same !

v i j 53·1 ( 3 s.f. )

3 443For we have

3

4

i j3 443v

3

4v3 4 i j

(a) magnitude 2 due east(b) magnitude 5 due south

Ans: 2i

Ans: 5 j

Any vector parallel to and can easily be written in the and form.

i ji j

Tell your partner what the following vectors would be in and form where and are unit vectors due east and due north respectively.

i j i j

j

i

We will see how to deal with vectors in other directions later.

The next 2 pages are needed only by those of you taking the MEI/OCR specification.

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Using and we can describe motion in 2-dimensions.

i j

If we want to predict the positions of two aircraft, for example, to make sure that they are not about to collide, we need 3-D.

jThe method used to write a quantity in 3-D is just an extension of that for 2-D.

ikx

y

z

e.g. The velocity of an aircraft is given by v3 4 2 i j k

Find the speed.Solution

: The magnitude of the vector gives the speed.

32 42 22

( You won’t be asked to find the direction.)

No minus sign

v 29 v v5·39 ( 3 s.f. )

The next section looks at Position Vectors and Displacement.

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Continue to Position Vectors and

Displacement

3. Position Vectors and Displacement

But, AO OBAB

If a body moves from A to B, gives the displacement of B from A.

AB

x

y

O

B

A

We often use single letters for position vectors, so

a

OBOA and b

and are called the position vectors of A and B.

OA OB

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

b

a

a b

a

AB ba

So, AB

Notation:The letter r is often used for the position vector of a body.

Rearranging to find the position vector of B gives

AB ba

So, may be given as

sr

Br A

Since displacement is given by s, we can have

r Br A

s

x

y

O

B

A

r A s

r B

or

x

y

O

B

A

b

a AB

Solution:

r A i3 2 ) kmj

Find the position vector of B.

y

xO

B

A

s

r Br A

s

The displacement of B from A is s

AB 6

j

r B

i3 2 )j 6

j r B

3 4 ) km

i j

e.g. The position vector of a point A is given by

A body moves from A to a point B where B is 6 km due north of A.

6 km

r B

r A

In the following, and are unit vectors east and north respectively.i j

r A i6 3 ) mjThe position vector of a point A is given

by

A body moves from A to a point B where B is 5 m due west of A. Find

EXERCISE

(a) the displacement of B from A, and(b) the position vector of

B.Solution

: s

AB(a)

r Br A

s r B

i6 3 )j

r B

3 ) m

i j

(b)

5 mi

5i

The next section looks at Column Vectors.

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Continue to Column Vectors

4. Column Vectors

Column vectors are just vectors written in columns !e.g. The position vectors of A and B are given by a

6 i jb

andi 4 2 j

Solution: a

42

b 6

1and

b aAB 6

1 42AB

AB

Write a and b as column vectors and find .

AB

Column vectors are just vectors written in columns !e.g. The position vectors of A and B are given by a

6 i jb

andi 4 2 j

Solution: a

42

b 6

1and

b aAB AB 6

1 42

1 4AB

Write a and b as column vectors and find .

AB

Column vectors are just vectors written in columns !e.g. The position vectors of A and B are given by a

6 i jb

andi 4 2 j

Solution: a

42

b 6

1and

b aAB AB 6

1 42

6 21 4AB

Write a and b as column vectors and find .

AB

Column vectors are just vectors written in columns !e.g. The position vectors of A and B are given by a

6 i jb

andi 4 2 j

Solution: a

Write a and b as column vectors and find .

AB

42

b 6

1and

b aAB AB 6

1 42

6 21 4AB

85

Tip: You don’t have to convert from and form to column vectors as either form can be used. However, students make fewer sign errors using column vectors.

i j

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This is the end of the final section of Vectors for Mechanics.