Www.le.ac.uk Vectors 3: Position Vectors and Scalar Products Department of Mathematics University of...

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Vectors 3:Position Vectors and Scalar ProductsDepartment of MathematicsUniversity of Leicester

Contents

Position Vectors

Introduction Introduction

Position Vector

Scalar Product

Introduction

• A point can be represented by a position vector that gives its distance and direction from the origin.

• At the point where two vectors meet, we can use an operation called the ‘scalar product’ to find the angle between them.

IntroductionPosition Vector

Scalar Product

Next

Position Vector

A

B

C

O

IntroductionPosition Vector

Scalar Product

Show position vector

corresponding to A.

Position Vector

A

B

C

Oa

IntroductionPosition Vector

Scalar Product

Show position vector

corresponding to B.

Position Vector

A

B

C

Oa

b

IntroductionPosition Vector

Scalar Product

Show position vector

corresponding to C.

Position Vector

A

B

C

Oa

c

b

IntroductionPosition Vector

Scalar Product

Next

What is the position vector of this point: ?

IntroductionPosition Vector

Scalar Product

5

3

5,3

yx 53

x

y

3

5

Vector between two points

x

y

O

A

B

b = OB

a = OAAB= b - a

If we start at the point A, travel along the vector a in the negative direction and then travel along the vector b; this is how we get the vector AB.

IntroductionPosition Vector

Scalar Product

Next

Click here to see this

illustrated.

x

y

O

A

B

b = OB

a = OA

x

y

O

A

B

b = OB

a = OAClick here to repeat.

a b

Click here to go back.

Distance between two points

x

y

O

A

B

b = OB

a = OAAB= b - a

A=

B=

The distance between A and B is the magnitude of AB

2

1

b

b

2

1

a

a

212

211 )()( ababAB

IntroductionPosition Vector

Scalar Product

Next

Questions:

IntroductionPosition Vector

Scalar Product

x

y

O

A

B

C

6

4

2

-2 2 4 6

What is the position vector BC? ( )What is the distance between A and B?

Next

Check Answers

Show Answers

Clear Answers

Scalar Product

• This is also known as dot product

• Takes two vectors of equal dimension and generates a scalar

332211

3

2

1

3

2

1

bababa

b

b

b

a

a

a

ba

IntroductionPosition Vector

Scalar Product

Next

Question...

2

3

2

1

IntroductionPosition Vector

Scalar Product

-1

-12

4

What is ?

Scalar Productcos332211 bababababa

x

y

a

bθ

IntroductionPosition Vector

Scalar Product

Next

cosbaba

Click here to see a proof of

.θ is the angle

between the two vectors, at the point where they meet.

It’s the smaller angle – not this one:

• Look at the triangle the vectors form:

a

bθ

b-a

We can use the Cosine Rule to find the angle θ in terms of a and b:

If we take and

we can evaluate...

cos2222 babaab

2

1

a

aa

2

1

b

bb

......continue

cos2222 babaab

......continue

cos2222 babaab

222cos2 abbaba

......continue

cos2222 babaab

222cos2 abbaba

23

22

21

23

22

21cos2 bbbaaaba

233

222

211 )()()( ababab

......continue

332211 222cos2 babababa

332211cos babababa

Make sure you multiply out the brackets yourself; you should get the following results:

Click here to go back

Scalar Product

cosbaba

cos332211 bababababa

To calculate the angle we rearrange the Scalar Product formula in the following way

baba1cos

IntroductionPosition Vector

Scalar Product

Next

2

-2

4

6

8

-6

-8

-4

0-2-6 -4 2 4 6 8-8

(Blue) (Pink)

y

x

IntroductionPosition Vector

Scalar Product

Next

.

Show angle between them

Clear

Draw vectors

Calculate angle

Question...

IntroductionPosition Vector

Scalar Product

Which of these angles is the angle calculated using ?

cosbaba

x

y

a

bθ

x

y

a

b

θx

y

a

bθ

x

y

a

b

θ

What is in the following diagram:

Question...

IntroductionPosition Vector

Scalar Product

Next

x

y

θ

3

1

1 3

53.13°

58.42°

78.46 °

What is if the two vectors are at right angles?

Question...

ba.

IntroductionPosition Vector

Scalar Product

0

1

ba

Question...

IntroductionPosition Vector

Scalar Product

6

-6

2

3

2

,

2

4

1

ba

What is ?ba.14

-14

An Interesting Dimension

0D

1D

2D

3D

4D

IntroductionPosition Vector

Scalar Product

Next

Conclusion

• Position Vectors are used to describe the size and position of a vector.

• Scalar Multiplication is used to find the angle between vectors.

• Two vectors are at right angles if and only if .

IntroductionPosition Vector

Scalar Product

Next

0. ba