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7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
1/18
7/27/2019 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
STROUD Worked examples and exercises are in the text
Introduction: scalar and vector quantities
Programme 6: Vectors
(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.
7/27/2019 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
STROUD Worked examples and exercises are in the text
Vector representatio n
Programme 6: Vectors
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
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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STROUD Worked examples and exercises are in the text
Vector representatio nTwo equal vectors
Types of vectors
Addition of vectors
The sum of a number of vectors
Programme 6: Vectors
STROUD Worked examples and exercises are in the text
Vector representatio n
Two equal vectors
Programme 6: Vectors
If two vectors, a and b , are said to be equal, they have the same magnitudeand the same direction
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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STROUD Worked examples and exercises are in the text
Vector representatio n
Programme 6: Vectors
If two vectors, a and b , have the same magnitude but opposite direction thena = b
STROUD Worked examples and exercises are in the text
Vector representatio n
Types of vectors
Programme 6: 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
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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STROUD Worked examples and exercises are in the text
Vector representatio n Addition of vectors
Programme 6: Vectors
The sum of two vectors and is defined as the single vector
STROUD Worked examples and exercises are in the text
Vector representatio n
The sum of a number of vectors
Programme 6: Vectors
Draw the vectors as a chain.
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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STROUD Worked examples and exercises are in the text
Vector representatio nThe sum of a number of vectors
Programme 6: Vectors
If the ends of the chain coincide the sum is 0.
STROUD Worked examples and exercises are in the text
Introduction: scalar and vector quantities
Vector 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
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
8/18
STROUD Worked examples and exercises are in the text
Components of a given vector
Programme 6: Vectors
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.
STROUD Worked examples and exercises are in the text
Components of a given vector
Components of a vector in terms of unit vectors
Programme 6: 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:
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
9/18
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
STROUD Worked examples and exercises are in the text
Vectors in space
Programme 6: Vectors
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:
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
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
STROUD Worked examples and exercises are in the text
Direction cosines
Programme 6: Vectors
The direction of a vector in three d imensions is determined by the angleswhich the vector makes with the three axes of reference:
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
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
STROUD Worked examples and exercises are in the text
Scalar product of two vectors
Programme 6: Vectors
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:
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
STROUD Worked examples and exercises are in the text
Scalar product of two vectors
Programme 6: Vectors
If a and b are two parallel vectors, the scalar product of a and b is then:
Therefore, given:
then:
STROUD Worked examples and exercises are in the text
Introduction: scalar and vector quantities
Vector 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
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
STROUD Worked examples and exercises are in the text
Vector product of two vectors
Programme 6: Vectors
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.
STROUD Worked examples and exercises are in the text
Vector product of two vectors
Programme 6: Vectors
If is a unit vector in the direction of:
then:
Notice that:
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
STROUD Worked examples and exercises are in the text
Vector product of two vectors
Programme 6: Vectors
Since the coordinate vectors are mutually perpendicular:
and
STROUD Worked examples and exercises are in the text
Vector product of two vectors
Programme 6: Vectors
So, given:
then:
That is:
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
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
STROUD Worked examples and exercises are in the text
Ang le b etween tw o vect or s
Programme 6: Vectors
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
7/27/2019 Mesin Fluida 1 Vektor+Segitiga Kecepatan
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1
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
STROUD Worked examples and exercises are in the text
Direction ratios
Programme 6: Vectors
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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