1. EE 543 Theory and Principles of Remote Sensing
Topic 1 – Review of Vector Calculus

2. Outline Vectors and vector addition Unit vectors
Ref:
Vectors and vector addition
Unit vectors
Base vectors and vector components
Rectangular coordinates in 2-D
Rectangular coordinates in 3-D
A vector connecting two points
Dot product
Cross product
Triple product
Triple vector product
Operators and Theorems
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3. Vector Definition
 A scalar is a quantity like mass or temperature that only has a magnitude.
On the other had, a vector is a mathematical object that has magnitude and direction, e.g. velocity, force.
A line of given length and pointing along a given direction, such as an arrow, is the typical representation of a vector.
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4. Vector Notation
Typical notation to designate a vector is a boldfaced character, a character with and arrow on it, or a character with a line under it (i.e., A, ).
The magnitude of a vector is its length and is normally denoted by
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5. Vector Addition
Addition of two vectors is accomplished by laying the vectors head to tail in sequence to create a triangle.
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6. Vector Algebra Rules P and Q are vectors and a is a scalar
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7. Unit Vectors A unit vector is a vector of unit length.
A unit vector is sometimes denoted by replacing the arrow on a vector with a "^" or just adding a "^" on a boldfaced character.
and
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8. Unit Vectors(2)
Any vector can be made into a unit vector by dividing it by its length.
Any vector can be fully represented by
providing its magnitude and a unit vector along
its direction.
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9. Base Vectors
The base vectors of a rectangular coordinate system are given by a set of three mutually orthogonal unit vectors denoted by and that are along the x, y, and z coordinate directions, respectively.
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10. Components Along Basis Vectors
In a rectangular coordinate system the components of the vector are the projections of the vector along the x, y, and z directions.
For example, in the figure the projections of vector A along the x, y, and z directions are given by Ax, Ay, and Az, respectively.
The magnitude can be calculated by
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11. Direction Cosines
The direction cosines can be calculated from the components of the vector and its magnitude through the relations
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12. Unit Vector Construction
A unit vector can be constructed along a vector using the direction cosines as its components along the x, y, and z directions:
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13. Vector Connecting Two Points
The vector connecting point A to point B is given by
A unit vector along the line A-B
can be obtained from
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14. Example 1 Addition of two vectors Add the two vectors:
What is the magnitude of the resulting vector?
What is its angle with respect to the x-axis?
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15. Solution 1 y A a x C B
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16. Example 2 Addition of three vectors: Add the vectors:
What is the magnitude of the resulting vector?
What is its angle with respect to the x-axis?
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17. Solution 2
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18. Example 3 Magnitude and angles of a vector
Find the magnitude and angles with respect of x, y and z axis of the vector:
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19. Solution 3
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20. Dot Product
The dot product is denoted by “.” between two vectors. The dot product of vectors A and B results in a scalar given by the relation
Order is not important in the dot product
Commutative
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21. Dot Product Properties
The angle between a vector and itself is zero. Thus:
Equals 1 when A = B
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22. Dot Product in Rectangular Coordinates
i, j, k are orthogonal vectors
i.i = j.j = k.k = 1
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23. Example 4 Dot Product Find the dot product of the two vectors:
What is the separation angle between A and B?
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24. Solution 4 =1
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25. Projection
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26. Cross Product (Vector Product)
The cross product of vectors a and b is a vector perpendicular to both a and b and has a magnitude equal to the area of the parallelogram generated from a and b.
The direction of the cross product is given by the right-hand rule . The cross product is denoted by a “X" between the vectors
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27. Right Hand Rule
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28. Cross Product (2)
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29. Cross Product(3) Order is important in the cross product.
If the order of operations changes in a cross product the direction of the resulting vector is reversed.
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30. Properties of Cross Product
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31. Cross Product in Rectangular Coordinatesz y x
Right Hand Rule
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32. Example
Find the cross product of the two vectors:
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33. Solution
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34. Solution 2*
* Valid only for Cartesian coordinates.
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35. The Triple Product The triple product has the following properties
 The triple product of vectors a, b, and c is given by and is a scalar quantity
The triple product has the following properties
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36. Triple Product in Rectangular Coordinates
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37. Triple Vector Product
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38. Vectors in Electromagnetics
In em, we typically deal with vectors that are functions of position for a given direction. Therefore, vector components along x, y and z are not constant.
The rate of change along a given direction is important in em. Electric and magnetic fields are related to each other through a differential operator.
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39. Main operators in vector calculus
Divergence
Gradient
Curl
Laplacian
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40. Vector Differentiation - Operator
One of the most important and useful mathematical constructs is the "del operator", usually denoted by (which is called the "nabla").
This can be regarded as a vector whose components in the three principle directions of a Cartesian coordinate system are partial differentiations with respect to those three directions.
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41. Operator
All the main operations of vector calculus, namely, the divergence, the gradient, the curl, and the Laplacian can be constructed from this single operator.
The entities on which we operate may be either scalar fields or vector fields.
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42. The Gradient (Scalar to vector)
If we simply multiply a scalar field such as p(x,y,z) by the del operator, the result is a vector field, and the components of the vector at each point are just the partial derivatives of the scalar field at that point, i.e.,
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43. The Divergence (Scalar Product, Dot Product) (Vector to scalar)
The divergence of a vector field v(x,y,z) = vx(x,y,z)i + vy(x,y,z)j + vz(x,y,z)k is a scalar
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44. The Curl (Vector Product, Cross Product) (Vector to vector)
The curl of a vector field v(x,y,z) = vx(x,y,z)i + vy(x,y,z)j + vz(x,y,z)k is a vector:
In Cartesian coordinates
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45. The Laplacian Operator (Scalar to scalar)
This is sometimes called the "div grad" of a scalar field, and is given by
For convenience we usually denote this operator by the symbol
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46. Stoke’s Theorem
The line integral of a vector along a closed path C is equal to the integral of the dot product of its curl and the normal to the surface which contains C as its contour. S ds dl C A
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47. Divergence Theorem
The dot product of a vector and the normal to a closed surface S is equal to the volume integral of its divergence over the volume that is contained by S.
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48. References
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