STROUD Worked examples and exercises are in the text Programme 6: Vectors VECTORS PROGRAMME 6

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Worked examples and exercises are in the text

Programme 6: Vectors

VECTORS

PROGRAMME 6

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Introduction: scalar and vector quantities

Vector representation

Components of a given vector

Vectors in space

Direction cosines

Scalar product of two vectors

Vector product of two vectors

Angle between two vectors

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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(a) A scalar quantity is defined completely by a single number with appropriate units

(b) A vector quantity is defined completely when we know not only its magnitude (with units) but also the direction in which it operates

Physical quantities can be divided into two main groups, scalar quantities and vector quantities.

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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A vector quantity can be represented graphically by a line, drawn so that:

(a) The length of the line denotes the magnitude of the quantity(b) The direction of the line (indicated by an arrowhead) denotes the

direction in which the vector quantity acts.

The vector quantity AB is referred to as or a

____AB

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Two equal vectors

Types of vectors

Addition of vectors

The sum of a number of vectors

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Two equal vectors

If two vectors, a and b, are said to be equal, they have the same magnitude and the same direction

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If two vectors, a and b, have the same magnitude but opposite direction then a = −b

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Types of vectors

(a) A position vector occurs when the point A is fixed

(b) A free vector is not restricted in any way. It is completely defined by its length and direction and can be drawn as any one of a set of equal length parallel lines

____AB

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Addition of vectors

The sum of two vectors and is defined as the single vector ____AB

____BC

____AC

____ ____ ____

AB BC AC

a b =c

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Draw the vectors as a chain.

____ ____ ____ ____ ____

____or

AB BC CD DE AE

a b c d

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If the ends of the chain coincide the sum is 0.

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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Just as can be replaced by so any single vector can be replaced by any number of component vectors so long as the form a chain beginning at P and ending at T.

____ ____ ____ ____AB BC CD DE

____AE

____PT

____PT a b c d

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Components of a vector in terms of unit vectors

The position vector , denoted by r can be defined by its two components in the Ox and Oy directions as:

____OP

(along O ) (along O )x y r a b

a b r i j

If we now define i and j to be unit vectors in the Ox and Oy directions respectively so that

then: and a b a i b j

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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In three dimensions a vector can be defined in terms of its components in the three spatial direction Ox, Oy and Oz as:

i is a unit vector in the Ox direction,j is a unit vector in the Oy direction and k is a unit vector in the Oz direction

Vectors in space

The magnitude of r can then be found from Pythagoras’ theorem to be:

2 2 2r a b c

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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Direction cosines

The direction of a vector in three dimensions is determined by the angles which the vector makes with the three axes of reference:

so that

cos therefore cos

a a rrb b rrc c rr

r i j k

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Direction cosines

Since:

2 2 2 2

2 2 2 2 2 2 2

2 2 2 1

= then

cos cos cos

a b c r

r r r r

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Direction cosines

Defining:

where [l, m, n] are called the direction cosines.

2 2 2 1l m n

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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If a and b are two vectors, the scalar product of a and b is defined to be the scalar (number):

where a and b are the magnitudes of the vectors and θ is the angle between them.

The scalar product (dot product) is denoted by:

cosab a.b

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If a and b are two parallel vectors, the scalar product of a and b is then:

Therefore, given:

1 2 3 1 2 3

1 1 2 2 3 3

a a a b b b

a b a b a b

a i j k b i j k

cos0ab ab a.b

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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The vector product (cross product) of a and b, denoted by:

is a vector with magnitude:

and a direction perpendicular to both a and b such that a, b and form a right-handed set.

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If is a unit vector in the direction of:

Notice that:

ˆsin ab a b n

b a a b

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Since the coordinate vectors are mutually perpendicular:

i j kj k ik i j

i i j j k k 0

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So, given:

That is:

1 2 3 1 2 3 and a a a b b b a i j k b i j k

2 3 3 2 1 3 3 1 1 2 2 1( ) ( ) ( )a b a b a b a b a b a b a b i j k

a a ab b b

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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Let a have direction cosines [l, m, n] and b have direction cosines [l′, m′, n′]

Let and be unit vectors parallel to a and b respectively.

therefore

2 2 2 2( ) ( ) ( ) ( )2 2( )2 2cos by the cosine rule

PP l l m m n nll mm nn

cos =ll mm nn

____OP

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Vectors in space

Direction cosines

Direction ratios

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Vectors in space

Direction cosines

Direction ratios

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Direction ratios

the components a, b and c are proportional to the direction cosines they are sometimes referred to as the direction ratios of the vector.

a b cl m nr r r

r i j k

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Programme 6: VectorsLearning outcomes

Define a vector

Represent a vector by a directed straight line

Add vectors

Write a vector in terms of component vectors

Write a vector in terms of component unit vectors

Set up a system for representing vectors

Obtain the direction cosines of a vector

Calculate the scalar product of two vectors

Calculate the vector product of two vectors

Determine the angle between two vectors

Evaluate the direction ratios of a vector

STROUD Worked examples and exercises are in the text Programme 6: Vectors VECTORS PROGRAMME 6

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