EE 543Theory and Principles of

Remote Sensing

Topic 1 – Review of Vector Calculus

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Outline

• 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

Ref: http://www.glenbrook.k12.il.us/GBSSCI/PHYS/Class/waves/u10l1b.html

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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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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 .

or A A

A

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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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Vector Algebra Rules

P and Q are vectors and a is a scalar

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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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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 alongits direction.

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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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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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Direction Cosines• The direction cosines can be calculated

from the components of the vector and its magnitude through the relations

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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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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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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?

ˆ ˆ6 3

ˆ ˆ4 7

A x y

B x y

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Solution 1

2 2

ˆ ˆ

ˆ ˆ10 4

10 4 116 10.8

ˆ 10cos 21.8

10

3

.

4 7

8

6

o

C

x y

C x y

C

BA

C x

C

x

y

A

B

C

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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?

ˆ ˆ8 12

ˆ ˆ5 15

ˆ ˆ7 9

A x y

B x y

C x y

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Solution 2

2 2

ˆ ˆ6 18

6 18 360 18.97

ˆ 6cos 108.4

18.97o

D x y

D

D x

D

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Example 3

Magnitude and angles of a vector

Find the magnitude and angles with respect of x, y and z axis of the vector:

ˆ ˆ ˆ10 10 10A x y z

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Solution 32 2 210 10 10 300 17.32

ˆ 10cos 0.577

17.32

ˆ 10cos 0.577

17.32

ˆ 10cos 0.577

17.32

54.7o

A

A x

A

A y

A

A z

A

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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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Dot Product Properties

The angle between a vector and itself is zero. Thus:

Equals 1 when A = B

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Dot Product in Rectangular Coordinates

i, j, k are orthogonal vectors

i.i = j.j = k.k = 1

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Example 4

Dot Product

Find the dot product of the two vectors:

ˆ ˆ3 2

ˆ ˆ5 8

A y z

B x y

What is the separation angle between A and B?

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Solution 4

2 2

2 2

1

ˆ ˆ ˆˆ3 2 5 8

ˆ ˆ ˆ ˆ ˆ ˆˆ ˆ3 5 3 8 2 5 2 8

24

cos( )

3 2 13

5 8 89

24cos 45.12

13 89

A B y z x y

y x y y z x z y

A B A B

A

B

=1

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Projection

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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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Right Hand Rule

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Cross Product (2)

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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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Properties of Cross Product

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Cross Product in Rectangular Coordinates

Right Hand Rule

xy

z

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Example

Find the cross product of the two vectors:

ˆ ˆ ˆ8 3 10

ˆ ˆ ˆ15 6 17

A x y z

B x y z

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Solution

ˆ ˆ ˆ ˆˆ ˆ8 3 10 15 6 17

ˆ ˆ ˆ ˆ8 6 8 17

ˆ ˆ ˆ ˆ3 15 3 17

ˆ ˆˆ ˆ10 15 10 6

ˆ ˆ ˆ111 14 93

A B x y z x y z

x y x z

y x y z

z x z y

x y z

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Solution 2*

* Valid only for Cartesian coordinates.

ˆ ˆ ˆ ˆˆ ˆ

det det 8 3 10

15 6 17x y z

x y z

x y z x y z

A B a a a

b b b

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The Triple Product

•  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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Triple Product in Rectangular Coordinates

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Triple Vector Product

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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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Main operators in vector calculus

• Divergence

• Gradient

• Curl

• Laplacian

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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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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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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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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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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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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 2

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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.

C S

A dl A ds

C

S

A

ds

dl

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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.

S V

A ds Adv

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References

1. http://em-ntserver.unl.edu/Math/mathweb/vectors/vectors.html

2. http://www.mathpages.com/home/kmath330/kmath330.htm