Today in Physics 217: vector analysis Vectors: ‰ have direction ‰ have magnitude Vector operations include ‰ vector addition ‰ vector multiplication by a scalar ‰ the dot product ‰ the cross product Vector components Vector transformation Second-rank tensors 4 September 2002

Physics 217, Fall 2002

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Vector operations Vector addition: ‰ Adding two vectors produces a third vector: A+B = C ‰ Vector addition is commutative: A+B =C = B+ A ‰ Vector subtraction is equivalent to adding the opposite of a vector:

A − B = A + ( −B ) 4 September 2002

Physics 217, Fall 2002

a)

B A

C=A+B

b)

C=B+A

A B

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Vector operations (continued) Vector multiplication by a scalar: ‰ The result of vector multiplication by a scalar is a vector. ‰ The magnitude of the resulting vector is the product of the magnitude of the scalar and the magnitude of the vector. ‰ The direction of the resulting vector is the same as the direction of the original vector if a > 0 and opposite to the direction of the original vector if a < 0. ‰ Vector multiplication is distributive: a ( A + B ) = aA + aB 4 September 2002

a)

Physics 217, Fall 2002

aA (a > 0) A

b) A

aA (a < 0)

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Vector operations (continued) The dot product (scalar product): ‰ The results of the dot product is a scalar:

A

A ⋅ B = A B cosθ = AB cos θ

‰ The dot product is commutative:

θ

A⋅B = B⋅ A

B

‰ The dot product is distributive: A ⋅(B + C) = A ⋅ B + A ⋅C 4 September 2002

Physics 217, Fall 2002

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Vector operations (continued) Cross product (vector product): ‰ The result of the cross product is a vector perpendicular to the two original vectors. • Magnitude: C = A × B = AB sin θ

C

• Direction: use right-hand rule ‰ The cross product is not commutative: A × B = −B × A

B θ A

‰ The cross product is distributive:

A× (B + C) = A× B + A×C 4 September 2002

Physics 217, Fall 2002

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Vector components A vector can be identified by specifying its three Cartesian components:

z axis A

A = Ax xˆ + Ay yˆ + Az zˆ Unit vectors Vector operations: ‰ To add vectors, add like components. ‰ To multiply a vector by a scalar, multiply each component. 4 September 2002

Physics 217, Fall 2002

y axis

x axis

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Vector components (continued) ‰ To calculate the dot product of two vectors, multiply like components and add: A ⋅ B = Ax Bx + Ay By + Az Bz

z axis

‰ To calculate the cross product of two vectors, evaluate the following determinant: xˆ

yˆ

zˆ

A × B = Ax

Ay

Az

Bx

By

Bz

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A

y axis

x axis

Physics 217, Fall 2002

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Vector transformation ‰ The components of a vector depend on the choice of the coordinate system. ‰ Different coordinate system will produce different components for the same vector. ‰ The choice of coordinate system being used can significantly change the complexity of problems in electrodynamics. 4 September 2002

z'

Physics 217, Fall 2002

z A θ' φ θ

y' y

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Vector transformation (continued) The coordinates of vector A in coordinate system S are related to the coordinates of vector A in coordinate system S’:  Ay′   cos φ   =   Az′   − sin φ

z'

sin φ   Ay    cos φ   Az 

The rotation considered here leaves the x axis untouched. The x coordinate of vector A will thus not change:

 Ax′   1 0     Ay′  =  0 cos φ    0 − sin φ A  z′   4 September 2002

z

0   Ax   I  sin φ   Ay  ≡ R ⋅ A cos φ   Az  Physics 217, Fall 2002

A θ' φ θ

y' y

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Vector transformation (continued) Coordinate transformation resulting from a rotation around an arbitrary axis can be written as:  Ax′   Rxx     Ay′  =  Ryx     Az′   Rzx

Rxy Ryy Rzy

Rxz   Ax   Rxx Ax + Rxy Ay + Rxz Az      Ryz   Ay  =  Ryx Ax + Ryy Ay + Ryz Az      Rzz   Az   Rzx Ax + Rzy Ay + Rzz Az 

or, more compactly, with x denoted as 1, y as 2, z as 3: Ai′ =

4 September 2002

3

∑ Rij A j j =1

Physics 217, Fall 2002

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Vector transformation (continued) The rotation matrix R is an example of a unitary transformation: one that does not change the magnitude of the object on which it operates: I A′ = R ⋅ A and A′ = A. If R is unitary,then

3

∑ Rij Rik = δ jk

,

i =1

 1 if j = k where δ jk =  0 otherwise

(the Kroneker delta),

as you will make plausible in this week’s homework. 4 September 2002

Physics 217, Fall 2002

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Second-rank tensors Vectors are first-rank tensors, having three independent components that can be represented by a column matrix. An I object T with nine independent components that can multiply a vector and produce a vector result, I B=T⋅A are called second-rank tensors. They behave as follows under rotations: 3 3 Tij′ = Rik R jlTkl

∑∑

k =1 l =1

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Physics 217, Fall 2002

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