Mesin Fluida 1 Vektor+Segitiga Kecepatan

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STROUD Worked examples and exercises are in the text

Introduction: scalar and vector quantitiesVector representatio n

Components of a given vector

Vectors in s pace

Direction cosines

Scalar product of two vectors

Vector product of two vectors

Ang le b etween tw o vect or s

Direction ratios

Programme 6: Vectors


Introduction: scalar and vector quantities


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

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

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


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Vectors in s pace

Direction cosines




Direction ratios



Vector representatio n


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 isreferred to as or a


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Vector representatio nTwo equal vectors

Types of vectors

Addition of vectors

The sum of a number of vectors




Two equal vectors


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


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



Types of vectors


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

(b) A line vector is such that it can slide along its line of action

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

parallel lines


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Vector representatio n Addition of vectors


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



The sum of a number of vectors


Draw the vectors as a chain.


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Vector representatio nThe sum of a number of vectors


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





Vectors in s pace

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.



Components of a vector in terms of unit vectors


The position vector , denoted by r can be defined by its two componentsin the O x and O y directions as:

If we now define i and j to beunit vectors in the O x and Oydirections respectively so that

then:


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Vectors in s pace

Direction cosines




Direction ratios



Vectors in space


In three dimensions a vector can be defined in terms of its components inthe three spatial direction O x, O y and O z as:

where k is a unit vector in the O z direction

The magnitude of r can then befound from Pythagoras theorem to

be:


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Vectors in s pace

Direction cosines




Direction ratios



Direction cosines


The direction of a vector in three d imensions is determined by the angleswhich the vector makes with the three axes of reference:


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Vectors in s pace

Direction cosines




Direction ratios





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

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

The scalar product ( dot product ) is denoted by:


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

Therefore, given:

then:





Vectors in s pace

Direction cosines




Direction ratios



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

is a vector with magnitude:

and a direction such that a , b andform a right-handed set.




If is a unit vector in the direction of:

then:

Notice that:


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Since the coordinate vectors are mutually perpendicular:

and




So, given:

then:

That is:


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Vectors in s pace

Direction cosines




Direction ratios





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


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Vectors in s pace

Direction cosines




Direction ratios



Direction ratios


Since

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


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Mesin Fluida 1 Vektor+Segitiga Kecepatan