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6161103 2.5 cartesian vectors

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6161103 2.5 cartesian vectors

  1. 1. 2.5 Cartesian VectorsRight-Handed Coordinate SystemA rectangular or Cartesian coordinatesystem is said to be right-handedprovided:- Thumb of right hand pointsin the direction of the positivez axis when the right-handfingers are curled about thisaxis and directed from thepositive x towards the positive y axis
  2. 2. 2.5 Cartesian VectorsRight-Handed Coordinate System- z-axis for the 2D problem would beperpendicular, directed out of the page.
  3. 3. 2.5 Cartesian VectorsRectangular Components of a Vector- A vector A may have one, two or threerectangular components along the x, y and zaxes, depending on orientation- By two successive application of theparallelogram law A = A’ + Az A’ = Ax + Ay- Combing the equations, A can beexpressed as A = Ax + Ay + Az
  4. 4. 2.5 Cartesian Vectors Unit Vector - Direction of A can be specified using a unit vector - Unit vector has a magnitude of 1 - If A is a vector having a magnitude of A ≠ 0, unit vector having the same direction as A is expressed by uA = A / ASo that A = A uA
  5. 5. 2.5 Cartesian VectorsUnit Vector- Since A is of a certain type, like forcevector, a proper set of units are used for thedescription- Magnitude A has the same sets of units,hence unit vector is dimensionless- A ( a positive scalar)defines magnitude of A- uA defines the directionand sense of A
  6. 6. 2.5 Cartesian VectorsCartesian Unit Vectors- Cartesian unit vectors, i, j and k are usedto designate the directions of z, y and z axes- Sense (or arrowhead) of thesevectors are described by a plusor minus sign (depending onpointing towards the positiveor negative axes)
  7. 7. 2.5 Cartesian VectorsCartesian Vector Representations- Three components of A act in the positive i,j and k directions A = Axi + Ayj + AZk*Note the magnitude anddirection of each componentsare separated, easing vectoralgebraic operations.
  8. 8. 2.5 Cartesian VectorsMagnitude of a Cartesian Vector- From the colored triangle, A = A2 + Az2- From the shaded triangle, A = Ax2 + Ay 2- Combining the equations givesmagnitude of A A = Ax2 + Ay + Az2 2
  9. 9. 2.5 Cartesian VectorsDirection of a Cartesian Vector- Orientation of A is defined as thecoordinate direction angles α, β and γmeasured between the tail of A and thepositive x, y and z axes- 0° ≤ α, β and γ ≤ 180 °
  10. 10. 2.5 Cartesian VectorsDirection of a Cartesian Vector- For angles α, β and γ (blue coloredtriangles), we calculate the directioncosines of A Ax cos α = A
  11. 11. 2.5 Cartesian VectorsDirection of a Cartesian Vector- For angles α, β and γ (blue coloredtriangles), we calculate the directioncosines of A Ay cos β = A
  12. 12. 2.5 Cartesian VectorsDirection of a Cartesian Vector- For angles α, β and γ (blue coloredtriangles), we calculate the directioncosines of A Az cos γ = A
  13. 13. 2.5 Cartesian VectorsDirection of a Cartesian Vector- Angles α, β and γ can be determined by theinverse cosines- Given A = Axi + Ayj + AZk- then, uA = A /A = (Ax/A)i + (Ay/A)j + (AZ/A)kwhere A = Ax2 + Ay + Az2 2
  14. 14. 2.5 Cartesian VectorsDirection of a Cartesian Vector- uA can also be expressed as uA = cosαi + cosβj + cosγk- Since A = Ax + Ay + Az2 and magnitude of uA 2 2= 1, cos 2 α + cos 2 β + cos 2 γ = 1- A as expressed in Cartesian vector form A = AuA = Acosαi + Acosβj + Acosγk = Axi + Ayj + AZk

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