© Boardworks Ltd 2008 1 of 45 S7 Vectors Maths Age 14-16

© Boardworks Ltd 2008 1 of 45

S7 Vectors

Maths Age 14-16


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S7.1 Vector notation

Contents

S7 Vectors

S7.5 Finding the magnitude of a vector

S7.6 Using vectors to solve problems

S7.4 Vector arithmetic

S7.3 Adding and subtracting vectors

S7.2 Multiplying vectors by scalars


Vectors and scalars

A vector is a quantity that has both size (or magnitude) and direction.A vector is a quantity that has both size (or magnitude) and direction.

Examples of vector quantities are:

A scalar is a quantity that has size (or magnitude) only.A scalar is a quantity that has size (or magnitude) only.

Examples of scalar quantities are:

velocity

displacement

force

speed

length

mass


Representing vectors

A vector can be represented using a line segment with an arrow on it.

For example,

A

B

The magnitude of the vector is given by the length of the line.

The direction of the vector is given by the arrow on the line.



A

B

This vector goes from the point A to the point B.

We can write this vector as AB.

Vectors can also be written using single letters in bold type.

For example, we can call this vector a.

When this is hand-written, the a is written as a

a



A

B

To go from the point A to the point B we must move 6 units to the right and 3 units up.

We can represent this movement using a column vector.

AB =63

This is the horizontal component. It tells us the number of units in the x-direction.This is the horizontal component. It tells us the number of units in the x-direction.

This is the vertical component. It tells us the number of units in the y-direction.This is the vertical component. It tells us the number of units in the y-direction.

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3




Equal vectors

Two vectors are equal if they have the same magnitude and direction.

All of the following vectors are equal:

They are the same length and parallel to each other.

a

bc

d

ef

g


The negative of a vector

Here is the vector AB =52

a

A

B

Suppose the arrow went in the opposite direction:

A

B

How can we describe this vector?

We can describe this vector as:

BA –a–5–2

or



a

A

B–a

A

B

If this is the vector a, this is the vector –a.

The negative of a vector is the same length and has the same slope, but goes in the opposite direction.

In general,

if a =xy

then –a =–x–y




The zero and unit vector

A vector with a magnitude of 0 is called the zero vector.

The zero vector is written as 0 or hand-written as 0

A vector with a magnitude of 1 is called a unit vector.

The most important unit vectors are those that are horizontal and vertical. These are called unit base vectors.

The horizontal unit base vector, , is called i.10

The vertical unit base vector, , is called j.01


The unit base vectors

The unit base vectors, i and j, can be represented in a diagram as follows:

j

i

Any column vector can easily be written in terms of i and j.

For example,5

–4= 5i – 4j

The number of i’s tells us how many units are moved horizontally and the number of j’s tell us how many units are moved vertically.


The unit base vectors

7i + 2j72

=

i – 3j1

–3=

–5i–50

=

Write the following in terms of unit base vectors.

Write the following in terms of column vectors.

41

4i + j =

0–7

–7j =

–18

–i + 8j =


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Contents

S7 Vectors







Multiplying vectors by scalars

Remember, a scalar quantity has size but not direction.

A scalar quantity can be represented by a single number.

A vector can be multiplied by a scalar. For example,

Suppose the vector a is represented as follows:

The vector 2a has the same direction but is twice as long.

a =32

2a =64

a 2a



In general, if the vector is multiplied by the scalar k, then xy

xy

k × =kxky

When a vector is multiplied by a scalar the resulting vector is either parallel to the original vector or lies on the same line.

Can you explain why this is?

For example, –25

3 × =–615




Pairs – parallel vectors


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Contents

S7 Vectors







Adding vectors

Adding two vectors is equivalent to applying one vector followed by the other. For example,

Suppose a =53

and b =3

–2

Find a + b

We can represent this addition in the following diagram:

a b

a + b

a + b =81


Adding vectors

When two or more vectors are added together the result is called the resultant vector.

In general, if a =ab

and b =cd

We can add two column vectors by adding the horizontal components together and adding the vertical components together.

a + b =a + cb + d


Adding vectors


Subtracting vectors

We can subtract two column vectors by subtracting the horizontal components and subtracting the vertical components. For example,

Find a – b

Suppose and b =–23

a =44

a – b =44

––23

=4 – –24 – 3

=61


Subtracting vectors

To show this subtraction in a diagram, we can think of a – b as a + (–b).

and b =–23

a =44

a b

a – b

a – b =61

–b a –b


Adding and subtracting vectors


The resultant vector


The parallelogram law for vector addition

As you have seen, we can use a parallelogram to demonstrate the addition of two vectors.

Suppose and b =23

a =4

–1

a

a + b

A

b

B

D

C

From this diagram we can see that

a + bAC = AB + BC =a + b

A

B

D

C

a

b Also

b + aAC = AD + DC =

a + b

A

B

D

Ca

b

Vector addition is commutativeVector addition is commutative


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Contents

S7 Vectors







Vector arithmetic

Suppose,

a =53

c =–14

andb =–26

Find

1) a + c a + c =53

+–14

=5 + –13 + 4

=47

2) 3b 3b = 3 ×–26

=3 × –23 × 6

=–618

3) 2c – b 2c – b = 2 ×–14

––26

=–2 – –2

8 – 6=

02


Vector equations

Suppose,

a =–47

b =6–1

and

Find vector c such that 2c + a = b

Start by rearranging the equation to make c the subject.

2c + a = b

2c = b – a

c = ½(b – a)


Vector equations

Suppose,

a =–47

b =6–1

and

Find vector c such that 2c + a = b

Next, substitute a and b to find c,

c = ½(b – a)

c = ½6 – –4–1 – 7

= ½ 10–8

c =5–4


A grid of congruent parallelograms


Vectors on a tangram

A tangram is an ancient Chinese puzzle in which a square ABCD is divided as follows:

A B

CD

F

E

G

H

I

J

Suppose,

AE = a and AF = b

Write the following in terms of a and b.

FC =

b

a

3b

HJ = –a

IG = a – 2b

CB = 2a – 4b

HI = b – a

HD = 2b – 3a


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Contents

S7 Vectors







Finding the magnitude of a vector


What is the magnitude of vector a?

We can find the magnitude using Pythagoras’ Theorem.

The direction of the vector is given by the arrow on the line.

a




What is the magnitude of vector a?

a

We often write |a| to represent the magnitude (or modulus) of a.

|a| = 32 + 52

= 34

= 5.83 (to 2 d.p.)

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5





Suppose, a =74

b =–52

and

Find

1) |a| |a| = 72 + 42 = 65 = 8.06 (to 2 d.p.)

2) |b| |b| = 52 + 22 = 29 = 5.39 (to 2 d.p.)

3) |a| + |b| |a| + |b| =65 + 29 = 13.45 (to 2 d.p.)

4) |a + b| a + b =26

|a + b| = 22 + 62 = 40 = 6.32 (to 2 d.p.)


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AS7.6 Using vectors to solve problems

Contents

S7 Vectors







Using vectors to solve problems

We can use vectors to solve many problems involving physical quantities such as force and velocity.

We can also use vectors to prove geometric results.

For example, suppose we have a triangle ABC as follows:

A

B

C

The line PQ is such that P is the mid-point of AB and Q is the mid-point of AC.

Use vectors to show that PQ is parallel to BC and that the length of BC is double the length of PQ.

P

Q



Let’s call vector a and vector b.

A

B

C

P

Q

a

b

AP AQ

PQ = –a + b

= b – a

BC = –2a + 2b

= 2b – 2a

= 2(b – a)

Therefore, BC = 2 PQ

We can conclude from this that PQ is parallel to BC and that the length of BC is double the length of PQ.



Suppose we have an object with two forces acting on it as shown:

6 N

8 N

Find the magnitude and direction of the resultant force.

The resultant force can be shown on the diagram as follows:

6 Nθ

8 N6 N

8 N



The magnitude of the resultant force can be found using Pythagoras’ theorem.

6 N

8 N

8 N6 N

θ

F2= 62 + 82

If F is the resultant force then

F2= 36 + 64

F2= 100

F = 10 N

F



The direction of the resultant force can be found using trigonometry.

6 N

8 N

8 N6 N

θ

If θ is the angle that the resultant force makes with the horizontal then,

tan θ =68

θ = 36 .87° (to 2 d.p.)

The resultant force has a magnitude of 10 N and makes an angle of 36.87° to the horizontal.

10 N

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© Boardworks Ltd 2008 1 of 45 S7 Vectors Maths Age 14-16