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Vector Algebra: Vectors vs. Scalars, Dot & Cross Products, and Coordinate Transformations , Study notes of Electrical and Electronics Engineering

An in-depth exploration of vectors and scalars, including their definitions, unit vectors, magnitudes, and vector algebra. It covers the concepts of dot product and cross product, as well as vector transformations in rectangular, cylindrical, and spherical coordinates. The document also includes formulas and explanations for coordinate systems, differential surfaces, and volumes.

Typology: Study notes

Pre 2010

Uploaded on 08/07/2009

koofers-user-ywt
koofers-user-ywt 🇺🇸

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Vectors vs. Scalars
A vector is a quantity having both magnitude and direction.
A scalar is a quantity only having magnitude.
A unit vector has magnitude one and is used to denote direction.
To find the unit vector of vector A:
222
zyx
A
zzyyxx
AAA
AAA
++
==
+
+
=
A
A
A
a
aaaA
The magnitude of vector A: 222
zyx AAA ++=A
pf3
pf4
pf5
pf8
pf9
pfa
pfd
pfe
pff
pf12

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Vectors vs. Scalars

A vector is a quantity having both magnitude and direction. A scalar is a quantity only having magnitude. A unit vector has magnitude one and is used to denote direction.

To find the unit vector of vector A:

2 2 2 x y z

A

x x y y z z

A A A

A A A

A

A

A

a

A a a a

The magnitude of vector A: A = Ax^2 + Ay^2 + A z^2

Vector Algebra

  • Vectors can be added and subtracted.
  • They obey associative,cummutative, and distributive laws
    • A+(B+C)=(A+B)+C
    • A+B=B+A
    • r(A+B)=rA+rB
  • Vector Multiplication has two forms:
    • Dot Product (scalar)
    • Cross Product (vector)

Projection

To find out how much of vector B is in the direction of a :

Cross Product

• The cross or vector product is:

A B B A

a a a A B

A B A B a

× =− ×

 × =

× =

x y z

x y z

x y z

AB n

B B B

A A A

sin θ

Expand this determinant

Rectangular Differential Surfaces

and Volumes

dydx a z

dx dz a y

dy dz a x

Volume = dx dy dz

Surfaces in Rectangular Coordinates

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

X

Y

Z

Surfaces in Cylindrical Coordinates

Spherical

dr

r sinθ dφ r dθ

r dθ dr aφφ

r sinθ dφ dr a θ

r sinθ dφ r dθ a r

dv= r sinθ dφ r dθ dr

Surfaces in Spherical Coordinates

Given:

A =Ax ax + Ay ay + Az az (rectangular)

Find:

A =Aρ aρρ +aφφ + Az az (cylindrical)

We must use the projection property

Aρ = A •aρρ

Aφ = A •aφφ

Az = A •az

Vector Transformation: Cylindrical

What is φ? use equations on page 17 #

Cylindrical Point transformations:

x= r cosφ

y= r sinφ

z= z

r = sqrt(x^2 +y^2 )

φ= tan-1^ y / x z= z

Note: A vector transformation is always done at a point

What is φ?

A = Ax ax + Ay ay + Az az

A = Ar ar + Aθ aθθ + Aφ aφφ

Ar = A• ar

Aθ = ΑΑaθθ

Aφ = Aaφφ

Then use table 1.2 for unit vector products.

!!For θ, φ use point transformations at some point (r, θ, φ)

Vector Transformations

Spherical: