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Vector Algebra

Vector Algebra. Definition A vector A has a magnitude A = | A | and a direction given by the unit vector â = A / | A |. In Cartesian coordinates ( x , y , z ), we find. Ulaby Figure 3-1. Ulaby Figure 3-2. Vector Algebra. Definition We also have. Ulaby Figure 3-1.

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Vector Algebra

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  1. Vector Algebra Definition A vector A has a magnitude A = |A| and a direction given by the unit vector â = A / |A|. In Cartesian coordinates (x, y, z), we find Ulaby Figure 3-1 Ulaby Figure 3-2

  2. Vector Algebra Definition We also have Ulaby Figure 3-1 Ulaby Figure 3-2

  3. Vector Algebra Equality Two vectors A and B are equal if Addition and Subtraction Ulaby Figure 3-3

  4. Vector Algebra Position and Distance vectors For a point P(x, y, z), the position vectorR goes from the origin to the point The distance vector between two points The distance d between two points equals |R12| Ulaby Figure 3-4

  5. Vector Algebra Multiplication — Simple ProductWhen a vector A is multiplied by a scalar k, the magnitude is multiplied by k and the direction is unchanged Multiplication — Scalar or Dot ProductThe scalar or dot product of two vectorsA and B is defined by where qAB is the angle between A and B When qAB < 90o, A·B > 0When qAB > 90o, A·B < 0 When qAB = 90o, A·B = 0, and the vectors are called orthogonal Ulaby Figure 3-5

  6. Vector Algebra Multiplication — Scalar or Dot ProductWe have so that Other properties: Ulaby Figure 3-5

  7. Vector Algebra Multiplication — Vector or Cross ProductThe vector or cross product of two vectorsA and B is defined by where qAB is the angle fromAtoBand is a unit vector determined by the right-hand rule The vector product is anti-commutative and distributive Ulaby Figure 3-6

  8. Vector Algebra Multiplication — Vector or Cross ProductWe have so that We may also write Ulaby Figure 3-6 This determinant form is very useful!

  9. Vector Algebra Summary of vector products

  10. Vector Algebra Ulaby Vectors and Angles: Ulaby Example 3-1 Question: In Cartesian coordinates, vector A is directed from the origin to the point P1(2, 3, 3), and vector B is directed from P1 to P2(1, –2, 2). Find (a) the vector A, its magnitude A, and its unit vector â, (b) the angle that A makes with the y-axis, (c) vector B, (d) the angle between A and B, and (e) the perpendicular distance from the origin to B. Answer: (a)

  11. Vector Algebra Ulaby Vectors and Angles: Ulaby Example 3-1 Answer (continued): (b) The angle b between A and the y-axis may be found from the expression (c) (d) (e) The points OP1P3 form a right triangle. The magnitude of the line segment OP3 is given by Ulaby Figure 3-7

  12. Vector Algebra Triple Scalar ProductThis product can be written in the equivalent forms The equivalence can be demonstrated from the determinant representation The absolute value of this scalar product is the volume of the parallelepiped whose sides are the vectors A, B, and C.

  13. Vector Algebra Triple Scalar ProductThis product can be written in the equivalent forms It corresponds to the volume of a parallelepiped B  C  A

  14. Vector Algebra Triple Vector ProductThis product is defined as This relationship can also be written in the form

  15. Orthogonal Coordinate Systems • Three coordinate systemsCoordinate systems (with three coordinates) specify points in space • Orthogonal systems have coordinates that are mutually perpendicular • — at least locally • There are three coordinate systems that are often used • Cartesian coordinates (x, y, z): Simplest and orthogonal everywhere • Cylindrical coordinates (r, q, z) • — used with optical fibers, coaxial cables, cylindrical waveguides • Spherical coordinates (R, q, f) • — used with antenna radiation, radar, earth-ionosphere waveguide

  16. Three Coordinate Systems Cartesian Coordinates Cylindrical Coordinates Spherical Coordinates Coordinate variables x, y, z r, q, z R, q, f Vector, A = Magnitude, |A| = Position vector Dot product, A·B Cross product, A×B Differential length, dl Differential surfaceareas Differential volume Summary of vector and differential relations

  17. Cartesian Coordinates • Differential Relations • For each coordinate system,there are three differentiallengths that also determinethe differential areas andvolumes Ulaby Figure 3-8

  18. Cylindrical Coordinates r = distance from the z-axis (0 < r < ) f = angle measured on the x-y plane from the +x-axis (0 < f < 2p) Ulaby Figure 3-9

  19. Cylindrical Coordinates y points in the direction of the maximum change in r; no change in f,z f = 2p / 3 f = p / 3 points in the direction of the maximum change in f; no change in r, z f = p / 6 f = 5p / 6 x f = p r = 1 r = 2 r = 3 r = 4 f = -p / 6 f = -5p / 6 f = -p / 3 f = -2p / 3 f = -p / 2

  20. Cylindrical Coordinates y points in the direction of the maximum change in r; no change in f,z x = 1 x = 3 x = -3 x = -1 y = 3 points in the direction of the maximum change in f; no change in r, z y = 1 x y = -1 y = -3

  21. Cylindrical Coordinates • Base vectors • Vector relations • Letting • We note that the direction of depends on f

  22. Cylindrical Coordinates Differential relations Ulaby Figure 3-10 Divergence

  23. Spherical Coordinates R = distance from the origin (0 < R < ) q = angle measured from the +z-axis (0 < q < p ) = zenith angle f = angle measured from the +x-axis on the x-y plane (0 < q < 2p ) = azimuthal angle Ulaby Figure 3-13

  24. Spherical Coordinates • Base vectors • Vector relations • Letting

  25. Spherical Coordinates Differential relations Ulaby Figure 3-14

  26. Spherical Coordinates Charge in a Sphere: Ulaby Example 3-6 Question: A sphere of radius 2 cm contains a charge of density rV given by What is the total charge? Answer: After converting from cm to m,

  27. Tech Brief 5: GPS • Originally developed by DOD • Originally 24 satellites, now 31 • Operation principles • Satellites constantly broadcast a message containing • Their location • Message time • System health and rough orbit details • GPS receivers use triangulation to determine location relative to satellites • Determine distances to each satellite by solving • an equation including the sat. position and • time, multiplying by the speed of light. Four satellites are needed to correct for the imprecise receiver clock (quartz).

  28. Tech Brief 5: Differential GPS • GPS accuracy: 20-30m • Differential GPS (DGPS) uses a static reference of known location in the receiver’s area to correct for inaccuracy factors • Time-delay errors (speed of light differences • Multipath interference • Satellite location errors • Reference receiver calculates correction • factors and transmits to DGPS receivers • in the area

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