Www.le.ac.uk Vectors 1: An Introduction to Vectors Department of Mathematics University of Leicester

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Vectors 1:An Introduction to Vectors

Department of MathematicsUniversity of Leicester

Contents

What is a Vector?

Dimensions

Basis

Drawing Vectors

What is a vector?

• A vector is a quantity which has direction as well as length

• They are used to determine the position of one point in space relative to another

What is a Vector? Dimensions Basis

Drawing Vectors

Next

Spot the DifferenceSo, what is the difference between a vector

and a scalar?

Next

What is a Vector? Dimensions Basis

Drawing Vectors

Spot the DifferenceSo, what is the difference between a vector

and a scalar?

• A scalar only has magnitude and is independent of direction

My car goes at 40mph.Scalar

man

Next

What is a Vector? Dimensions Basis

Drawing Vectors

Spot the DifferenceSo, what is the difference between a vector

and a scalar?

• A scalar only has magnitude and is independent of direction

• A vector has both magnitude AND direction

I’m going 20 miles West.

Vector man

My car goes at 40mph.Scalar

man

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What is a Vector? Dimensions Basis

Drawing Vectors

Spot the Difference

• Do we use a scalar or vector quantity to represent the top speed of a car?

Next

What is a Vector? Dimensions Basis

Drawing Vectors

Spot the Difference

• Do we use a scalar or vector quantity to represent the top speed of a car?

A: We use a SCALAR ie. the top speed of my car is 120mph. However, if I were to talk about the velocity of the car, this concerns both the speed AND direction in which I am travelling, so a vector is used for velocity.

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What is a Vector? Dimensions Basis

Drawing Vectors

Question…Choose whether the following would be measured using vectors or scalars:

Temperature

Position of the Earth in it’s orbit around the sun.

Distance travelled on a circular walk.

Next

What is a Vector? Dimensions Basis

Drawing Vectors

ScalarVector

ScalarVector

ScalarVector

Check Answers Clear Answers Show Answers

Dimensions

• The formal definition of dimension is the minimum number of coordinates used to specify a point in that dimension.

• A LINE is one dimensional; we only need a single number to get to any point along it (think about the number line).

• A PLANE or SURFACE is two dimensional; just think about a graph. We need the value along the x and y axis to find a point. Next

What is a Vector? Dimensions Basis

Drawing Vectors

Dimensions

• We live in a three dimensional world; can you give a reason why?

Next

What is a Vector? Dimensions Basis

Drawing Vectors

Basis

A basis is a set of linearly independent vectors which spans a space. This means any point in the space can be written as a combination of scalar multiples of the basis vectors.

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What is a Vector? Dimensions Basis

Drawing Vectors

Basis Vectors

Let’s think about the basis vectors of the 2D space.

Any point in this space can be described as the vector .

We can also write this in the following way:

y

x

1

0

0

1yx

y

x

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What is a Vector? Dimensions Basis

Drawing Vectors

Basis Vectors

• We say the vectors and

are basis vectors of the 2D space.

• In a similar way, , ,

are basis vectors of the 3D space.

• e1, e2 and e3 are also called i, j and k.

0

11e

1

02e

0

0

1

1e

0

1

0

2e

1

0

0

3e

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What is a Vector? Dimensions Basis

Drawing Vectors

Question….

Which of the following is equivalent to ?

kji 42

12

4

kji 42

kji 24

What is a Vector? Dimensions Basis

Drawing Vectors

( )

Vector in two dimensions

x

yab

v

v =

v = i+ ja b

Here we find the position relative to the origin

Click here to see how the vector

is formed.

Next

What is a Vector? Dimensions Basis

Drawing Vectors

x

y bjaib

av

x

y

v

a

b Click here to Repeat

Click here to Go Back

bjaib

av

Try drawing some vectors

x

y

v =

v = i + j

2

-2

4

6

8

-6

-8

-4

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

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What is a Vector? Dimensions Basis

Drawing Vectors

Draw this

Draw this

Clear

x

z abc

v =

yv

v = i+ j+ k

a b c

Vector in three dimensionsClick here to see how

the vector is formed.

Next

What is a Vector? Dimensions Basis

Drawing Vectors

x

z

y

ckbjai

c

b

a

v

x

z

yv

Click to here repeat

Click here to go back

ckbjai

c

b

a

v

a b

c

kji 5 kji 5kji 5

Which of the following is equivalent to this vector:

x

z

y

v5

1

1

What is a Vector? Dimensions Basis

Drawing Vectors

Conclusion

Vectors are quantities that have a size and a direction.

They can be in any number of dimensions.

The number of basis vectors required is equal to the number of dimensions.

Next

What is a Vector? Dimensions Basis

Drawing Vectors