Vector_Notes_-_All - EM Research Group Vector Primer...

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EM Research Group Vector Primer Prepared by the Electromagnetic Research Group Faculty Dennis Nyquist Edward Rothwell Leo Kempel Shanker Balasubramaniam March 23, 2004
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EM Research Group Scalar vs. Vector Quantities Ö Defined by magnitude and direction Ö Must define a set of laws analogous to those for scalar quantities Ö Examples: Force, velocity, electric field, and magnetic field Ö Completely specified by a single number along with a given dimensional unit Ö Obeys usual algebraic laws (e.g. addition, subtraction, multiplication, and division) Ö Examples: Mass and charge 1 P 2 P 12 12 d d r r = Note: Sometimes vectors are denoted by an arrow above the symbol, sometimes by a line, and sometimes with a bold font. We will use both for presentation clarity.
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EM Research Group Unit Vectors and Vector Length Definition: The length of a vector, its magnitude, is designated as: A = A r bold normal font A A r r = a ˆ Definition: A unit vector has a length of unity: Note: Unit vectors point in the direction of positive increase for the corresponding coordinate . Definition: Product of a vector A and a scalar, φ , is a vector having the direction of the vector, but the magnitude is φ times the original length a ˆ A A P r r r φ = φ = Note: Any vector may be expressed as the product of its length with its unit vector.
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EM Research Group A Comment on Notation This is what we will use… Various notation conventions exist… A A r r = = = A A ( )( ) () φ θ φ θ φ ρ φ ρ ˆ , ˆ , r ˆ | , , r z ˆ , ˆ , ˆ | z , , z ˆ , y ˆ , x ˆ | z , y , x Cartesian Cylindrical Vectors Spherical x x 1 ˆ i ˆ a ˆ x ˆ = = = Unit vectors A r = A On the board In the notes Three most common coordinate systems… z , y , x ( ) z , , z , , r φ ρ = φ ( ) ( ) φ θ = φ θ , , r , , R Cartesian Cylindrical Spherical
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EM Research Group Cartesian Components x y z x ˆ y ˆ z ˆ y A x A z A z y x A z ˆ A y ˆ A x ˆ + + = A r Decomposition of a vector into Cartesian components A r = x ˆ A x A r = y ˆ A y A r = z ˆ A z 2 z 2 y 2 x A A A + + = A r
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EM Research Group Addition of Vectors z y x A z ˆ A y ˆ A x ˆ + + = A r z y x B z ˆ B y ˆ B x ˆ + + = B r () ( ) ( ) z z y y x x B A z ˆ B A y ˆ B A x ˆ + + + + + = + = B A C r r r A B C y x A B B A v r r r + = + Commutative Law: ( ) ( ) C B A C B A r r r r r r + + = + + Associative Law: Note: All addition is done component-wise.
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EM Research Group Subtraction of Vectors z y x A z ˆ A y ˆ A x ˆ + + = A r z y x B z ˆ B y ˆ B x ˆ + + = B r () ( ) ( ) z z y y x x B A z ˆ B A y ˆ B A x ˆ + + = = B A C r r r Definition: The negative of a vector has the same length as the original vector but pointing in the opposing direction. C -B A y x Note: All subtraction is done component-wise.
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EM Research Group Dot (or Scalar) Product z y x A z ˆ A y ˆ A x ˆ + + = A r z y x B z ˆ B y ˆ B x ˆ + + = B r ( ) θ = + + = cos B A B A B A z z y y x x B A B A r r r r A few useful facts… ( ) θ cos B A r r θ B r A r 0 y ˆ z ˆ x ˆ z ˆ 0 z ˆ y ˆ x ˆ y ˆ 0 z ˆ x ˆ y ˆ x ˆ 1 z ˆ z ˆ 1 y ˆ y ˆ 1 x ˆ x ˆ = = = = = = = = = 2 z 2 y 2 x 2 A A A + + = = A A A r r r A Special Case
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EM Research Group Projection of One Vector onto Another
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This note was uploaded on 01/17/2011 for the course ECE 305 taught by Professor Staff during the Summer '08 term at Michigan State University.

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Vector_Notes_-_All - EM Research Group Vector Primer...

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