Difference between revisions of "Forest UCM NLM"

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<math>\vec{a} \times \left ( \vec{b} + \vec{c} \right ) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}</math>
 
<math>\vec{a} \times \left ( \vec{b} + \vec{c} \right ) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}</math>
  
=== The scaler triple product ===
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=== The scalar triple product ===
  
 
;definition
 
;definition
:scaler tiple product <math>\equiv \vec{a} \cdot \left (\vec{b} \times  \vec{c} \right )</math>
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:scalar triple product <math>\equiv \vec{a} \cdot \left (\vec{b} \times  \vec{c} \right )</math>
  
  

Revision as of 03:42, 7 August 2014


Newton's Laws of Motion

Limits of Classical Mechanic

Classical Mechanics is the formulations of physics developed by Newton (1642-1727), Lagrange(1736-1813), and Hamilton(1805-1865).

It may be used to describe the motion of objects which are not moving at high speeds (0.1[math] c[/math]) nor are microscopically small ( [math]10^{-9} m[/math]).

The laws are formulated in terms of space, time, mass, and force:


Vectors

Vector Notation

A vector is a mathematical construct of ordered elements that represent magnitude and direction simultaneously.

[math]\vec{r} = x \hat{i} + y \hat{j} + z \hat{k} = (x,y,z) = \sum_1^3 r_i \hat{e}_i[/math]


Vectors satisfy the commutative (order of addition doesn't matter) and associative ( doesn't matter which you add first) properties.


The multiplication of two vectors is not uniquely defined. At least three types of vector products may be defined.

Scalar ( Dot ) product

definition

[math]\vec{a} \cdot \vec{b} = \left | a \right | \left | b \right | cos \theta = a_1 b_1 + a_2 b_2 + a_3 b_3[/math]

physical intepretation
[math]\frac{\vec{a} \cdot \vec{b}}{\left | \vec{b} \right |}[/math] is the length of [math]\vec{a}[/math] that is along the direction of [math]\vec{b}[/math] (a projection like the casting of a shadow)

Commutative property of scalar product

[math]\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} [/math]

proof
[math]\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3[/math] definition of dot product
[math] a_1 b_1 + a_2 b_2 + a_3 b_3=b_1 a_1 + b_2 a_2 + b_3 a_3 [/math] comutative property of multiplication
[math] b_1 a_1 + b_2 a_2 + b_3 a_3=\vec{b} \cdot \vec{a}[/math] definition of dot product
[math]\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}[/math]

Distributive property of scalar product

[math]\vec{a} \cdot \left ( \vec{b} + \vec{c} \right ) = \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c}[/math]

Vector ( Cross ) product

definition
[math]\vec{a} \times \vec{b} = \left( a_2b_3-a_3b_2\right) \hat{e}_1 +\left( a_3b_1-a_1b_3\right) \hat{e}_2 +\left( a_1b_2-a_2b_1\right) \hat{e}_3[/math]

The vector product of [math]\vec{a}[/math] and [math]\vec{b}[/math] is a third vector [math]\vec{c}[/math] with the following properties.

[math]\left | \vec{c} \right | = \left | \vec{a} \right | \left | \vec{b} \right | \sin \theta[/math]
[math]\vec{c}[/math] is [math]\perp[/math] to [math]\vec{a}[/math] and [math]\vec{b}[/math]
the right hand rule convention is used to determine the direction of [math]\vec{c}[/math]
physical interpretation
[math]A = \left | \vec{a} \times \vec{b} \right | =[/math] area of a parallelogram with vectors [math]\vec{a}[/math] and [math]\vec{b}[/math] forming adjacent edges

let [math]h[/math] represent the perpendicular distance from the teminus of [math]\vec{b}[/math] to the line of action of [math]\vec{a}[/math] ( a.k.a. the height)

then the area of the parallelogram is given by

[math]A=\left | \vec{a} \right | h[/math]

the height [math]h[/math] is equivalent to [math]\left | \vec{b} \right | \sin \theta[/math] where [math]\theta[/math] is the angle between the vectors [math]\vec{a}[/math] and [math]\vec{b}[/math]

thus

[math]A=\left | \vec{a} \right | h = \left | \vec{a} \right | \left ( \left | \vec{b} \right | \sin \theta \right ) = \left | \vec{a} \times \vec{b} \right | [/math]

NON-Commutative property of vector product

[math]\vec{a} \times \vec{b} = -\vec{b} \times \vec{a} [/math]

proof
[math]\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3[/math] definition of dot product
[math] a_1 b_1 + a_2 b_2 + a_3 b_3=b_1 a_1 + b_2 a_2 + b_3 a_3 [/math] comutative property of multiplication
[math] b_1 a_1 + b_2 a_2 + b_3 a_3=\vec{b} \cdot \vec{a}[/math] definition of dot product
[math]\vec{a} \cdot \vec{b}=\vec{b} \cdot \vec{a}[/math]

Distributive property of the vector product

[math]\vec{a} \times \left ( \vec{b} + \vec{c} \right ) = \vec{a} \times \vec{b} + \vec{a} \times \vec{c}[/math]

The scalar triple product

definition
scalar triple product [math]\equiv \vec{a} \cdot \left (\vec{b} \times \vec{c} \right )[/math]


A third vector product is the tensor direct product.

Space and Time

Space

Cartesian, Spherical, and Cylindrical coordinate systems are commonly used to describe three-dimensional space.

Forest_UCM_NLM_Ch1_CoordSys



Forest_Ugrad_ClassicalMechanics