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FACULTY OF ENGINEERING COMPUTING AND MATHEMATICS

 © 2012 SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING

CITS2401

Computer Analysis and Visualisation

FACULTY OF ENGINEERING COMPUTING AND MATHEMATICS

 © 2012-2013 SCHOOL OF COMPUTER SCIENCE AND SOFTWARE ENGINEERING

CITS2401

Computer Analysis and Visualisation

Dot Products

Week 7

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Dot Products

The dot product

is sometimes

called the scalar

product

the sum of theresults when you

multiply two

vectors together,

element byelement.

Equivalent

statements

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Example Calculating the Centre of mass(centre of gravity)

Finding the centre of mass of a structure is important in a

number of engineering applications

The location of the center of mass can be calculated by

dividing the system up into small components.

 xm =  x 1m

1+  x 

2m

2 +  x 

3m

3+ etc...

 ym =   y1m1 +  y2m2  +  y3m3 + etc...

 zm =  z1m

1+  z

2m

2 +  z

3m

3+ etc...

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In a rectangular coordinate system

• are the coordinates of the center of mass• m is the total mass of the system

• x1, x2, and x3 etc are the x coordinates of each systemcomponent

• y1, y2, and y3 etc are the y coordinates of each systemcomponent

• z1, z2, and z3 etc are the z coordinates of each systemcomponent

• m1, m2, and m3 etc are the mass of each system component

 x , y, and  z

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Example Calculating the Center of mass

1D two mass example

 x   =  x 1m1 +  x 2m2

m

 x=0

 x=x1

 x=x2

m2m1

m =m1+m

2where

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In this example…

We’ll find the center of mass of a small collection of thecomponents used in a complex space vehicle

Item x, meters y, meters z meters Mass

Bolt 0.1 2 3 3.50 gram

screw 1 1 1 1.50 gram

nut 1.5 0.2 0.5 0.79 gram

bracket 2 2 4 1.75 gram

Formulate the problem using a dot product

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Input and Output

Input

• Location of each component in anx-y-z coordinate system – in meters

• Mass of each component, in grams

Output

• Location of the centre of mass

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Hand ExampleFind the x coordinate of the centre of mass

Item x, meters Mass,

gram

x * m, gram

meters

Bolt 0.1 x 3.50 = 0.35

screw 1 x 1.50 = 1.50

nut 1.5 x 0.79 = 1.1850

bracket 2 x 1.75 = 3.5

sum 7.54 6.535

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We know that…

The x coordinate is equal to

So…

=6.535/7.54 = 0.8667 meters

 x   =

 xi

i=1

4

∑   mi

mTotal

=

 xi

i=1

4

∑   mi

mi

i=1

4

∑

 x 

This is a dot

product

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We could use a plot to evaluate ourresults

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