> 7 0w9bjbjUU 177x$l99989~:cf2^;^;";;;;;;ddddddd$g ide;;;;;ehC;;fhChChC;;;dhC;dhCbhCCRcd;R;@{]869fAdd3f0cf djhCjdhCLecture No. 4
Subject: Cartesian Vectors
Objectives of Lecture:
To express forces as Cartesian vectors in three dimensions.
To determine magnitude and direction of a Cartesian vector.
To carry the addition and subtraction of Cartesian vector
4.1 RightHanded Coordinate System (i.e. 3D Coordinate System)
A rectangular or Cartesian coordinate system is said to be righthanded provided the thumb of the right hand points in the direction of the positive z axis when the right hand fingers are curled about this axis and directed from the positive x toward the positive y axis, as shown in Fig.4.1.
Fig.4.1: Right handed coordinate system
Rectangular Components of a Vector
A vector A may have one, two, or three rectangular components along the x, y, z coordinate axes, depending on how the vector is oriented relative to the axes.
For example, considering the vector A directed within an octant of the x, y, z frame, as shown in Fig.4.2.
Fig.4.2
The vector A may be resolved into components Ax, Ay, and Az by two successive applications of the parallelogram law, as follows:
A = A2 + Az and A2 = Ax + Ay
Therefore A = Ax + Ay + Az
Unit Vector
A unit vector, u, is used to specify the direction of a vector. The unit vector has a magnitude of 1 and it is a dimensionless vector.
If A is a vector having a magnitude A `" 0, then the unit vector having the same direction as A is represented by
uA = A / A
So that
A = A uA
Cartesian Unit Vectors
In three dimensions, the set of Cartesian unit vectors, i, j, k, is used to designate the direction of the x, y, z axes respectively. The positive Cartesian unit vectors are shown in Fig.4.3.
Fig.4.3
A unit vector having direction opposite to the direction as shown in Fig.4.3 will be expressed with negative sign.
Cartesian Vector Representation
A vector A as shown in Fig.4.4 may be expressed as Cartesian vector, as follows:
Fig.4.4
A = Axi + Ayj + Azk
uA = (Ax /A) i + (Ay /A) j + (Az /A) k
Magnitude of a Cartesian Vector
The magnitude of a Cartesian vector, A = Axi + Ayj + Azk, may be be obtained as follows:
Fig.4.5
Referring to the Fig.4.5 we can write
A = ( A2 2 + A2z )0.5 and A2 = ( Ax2 + A2y )0.5
Therefore A = (A2x + A2y + A2z)0
Direction of a Cartesian Vector
The direction, i.e. orientation, of a vector A is defined by the coordinate direction angles (alpha) ( beta ), and ( gamma), measured between the tail of A and the positive x, y, z axes located at the tail of A, as shown in Fig.4.6.
Fig.4.6
Cosines of the coordinate direction angles are given as:
cos = Ax / A cos = Ay / A cos = Az / A
Note:
Regardless of where A is directed each of the coordinate direction angles will be between 0 and 180.
Coordinate direction angle from positive axis + Coordinate direction angle from negative axis = 180.
The important relation between the direction cosines can be formulated as:
cos2 + cos2 +cos2 = 1
The vector A may be expressed in terms of the cosines of the coordinate direction angles as:
A = A cos i + A cos j + A cos k
Addition and Subtraction of Cartesian Vectors
Two Cartesian vectors, A = Axi + Ayj + Azk and B = Bxi + Byj + Bzk, as shown in Fig.4.7
Fig.4.7
may be added to form a resultant Cartesian vector R as:
R = A + B = ( Ax + Bx)i + ( Ay + By)j +( Az + Bz)k
may be subtracted to form a resultant Cartesian vector R2 as:
R2 = A  B = ( Ax  Bx)i + ( Ay  By)j +( Az  Bz)k
The above vector addition may be generalized for several concurrent forces as:
FR = F = Fxi + Fyj + Fzk
Where Fx, Fy , and Fz represent the algebraic sums of the respective x, y, z or i,j,k components of each force in the system.
Numerical Examples
Try the solved examples from Book (Examples 2 8 to 2 11, page 47 to 50)
Unsolved Examples
Determine the magnitude and coordinate direction angles of F1 = { 60i 50j + 40k } and F2= { 40i 85j + 30k }N. Sketch each force on an x,y,z reference.
Express each force in Cartesian vector form ( Fig.4.8 )
Fig.4.8
Determine the magnitude and the coordinate direction angles of the resultant force ( Fig.4.9 )
Fig.4.9
44 The joint, as shown in Fig.4.10, is subjected to the three forces. Express each force in Cartesian vector form and determine the magnitude and direction angles of the resultant force.
Fig.4.10
Try the other unsolved examples from Book
(Examples 2 64 to 2 80, page 52 to 54)
Lecture No. 4
Subject: Cartesian Vectors
4.0 Objectives of Lecture:
To express forces as Cartesian vectors in three dimensions.
To determine magnitude and direction of a Cartesian vector.
To carry the addition and subtraction of Cartesian vector
RightHanded Coordinate System
(i.e. 3D Coordinate System)
A rectangular or Cartesian coordinate system is said to be righthanded provided the thumb of the right hand points in the direction of the positive z axis when the right hand fingers are curled about this axis and directed from the positive x toward the positive y axis, as shown in Fig.4.1.
Fig.4.1: Right handed coordinate system
Rectangular Components of a Vector
A vector A may have one, two, or three rectangular components along the x, y, z coordinate axes, depending on how the vector is oriented relative to the axes.
For example, considering the vector A directed within an octant of the x, y, z frame, as shown in Fig.4.2.
Fig.4.2
The vector A may be resolved into components Ax, Ay, and Az by two successive applications of the parallelogram law, as follows:
A = A2 + Az and A2 = Ax + Ay
Therefore A = Ax + Ay + Az
43 Unit Vector
A unit vector, u, is used to specify the direction of a vector. The unit vector has a magnitude of 1 and it is a dimensionless vector.
If A is a vector having a magnitude A `" 0, then the unit vector having the same direction as A is represented by
uA = A / A
So that
A = A uA
44 Cartesian Unit Vectors
In three dimensions, the set of Cartesian unit vectors, i, j, k, is used to designate the direction of the x, y, z axes respectively. The positive Cartesian unit vectors are shown in Fig.4.3.
Fig.4.3
A unit vector having direction opposite to the direction as shown in Fig.4.3 will be expressed with negative sign.
Cartesian Vector Representation
A vector A as shown in Fig.4.4 may be expressed as Cartesian vector, as follows:
Fig.4.4
A = Axi + Ayj + Azk
uA = (Ax /A) i + (Ay /A) j + (Az /A) k
Magnitude of a Cartesian Vector
The magnitude of a Cartesian vector, A = Axi + Ayj + Azk, may be be obtained as follows:
Fig.4.5
Referring to the Fig.4.5 we can write
A = ( A2 2 + A2z )0.5 and A2 = ( Ax2 + A2y )0.5
Therefore A = (A2x + A2y + A2z)0
Direction of a Cartesian Vector
The direction, i.e. orientation, of a vector A is defined by the coordinate direction angles (alpha) ( beta ), and ( gamma), measured between the tail of A and the positive x, y, z axes located at the tail of A, as shown in Fig.4.6.
Fig.4.6
Cosines of the coordinate direction angles are given as:
cos = Ax / A cos = Ay / A cos = Az / A
Note:
Regardless of where A is directed each of the coordinate direction angles will be between 0 and 180.
Coordinate direction angle from positive axis + Coordinate direction angle from negative axis = 180.
The important relation between the direction cosines can be formulated as:
cos2 + cos2 +cos2 = 1
The vector A may be expressed in terms of the cosines of the coordinate direction angles as:
A = A cos i + A cos j + A cos k
Addition and Subtraction of Cartesian Vectors
Two Cartesian vectors, A = Axi + Ayj + Azk and B = Bxi + Byj + Bzk, as shown in Fig.4.7
Fig.4.7
may be added to form a resultant Cartesian vector R as:
R = A + B = ( Ax + Bx)i + ( Ay + By)j +( Az + Bz)k
may be subtracted to form a resultant Cartesian vector R2 as:
R2 = A  B = ( Ax  Bx)i + ( Ay  By)j +( Az  Bz)k
The above vector addition may be generalized for several concurrent forces as:
FR = F = Fxi + Fyj + Fzk
Where Fx, Fy , and Fz represent the algebraic sums of the respective x, y, z or i,j,k components of each force in the system.
Numerical Examples
Try the solved examples from Book
(Examples 2 8 to 2 11, page 47 to 50)
Unsolved Examples
Determine the magnitude and coordinate direction angles of F1 = { 60i 50j + 40k } and F2= { 40i 85j + 30k }N. Sketch each force on an x,y,z reference.
Express each force in Cartesian vector form ( Fig.4.8 )
Fig.4.8
Determine the magnitude and the coordinate direction angles of the resultant force ( Fig.4.9 )
Fig.4.9
44 The joint, as shown in Fig.4.10, is subjected to the three forces. Express each force in Cartesian vector form and determine the magnitude and direction angles of the resultant force.
Fig.4.10
Try the other unsolved examples from Book
(Examples 2 64 to 2 80, page 52 to 54)
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