defense technical paper

8/14/2019 Defense Technical Paper

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Sharyn D. Villarez

Jann Kebby G. Quilao

Elynah Roseyell M. Mistiola

Christine N. Guevarra

Jholo J. Zulueta

October 2013


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There are set of useful methods that can be used to

find the area of the triangle but these methods could risk

the definitive answer to the problem. In order to obtain a

definite answer, the method to be used must be

meticulously examined and ensured to produce an exact

outcome, but due to the existence and possibility of

mathematical errors, absolute values are taken under

consideration. The analysis of two methods, the Matrix and

Herons Formula, aimed to determine the more convenientand distinctive way to find the area given the coordinates of

a triangle.


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In mathematics, a matrix (plural matrices) isa rectangular array of numbers, symbols, or expressions,arranged in rows and columns. The individual items in amatrix are called its elements or entries.

A matrix is a concise and useful way of uniquelyrepresenting and working with linear transformations. Inparticular, every linear transformation can berepresented by a matrix, and every matrix correspondsto a unique linear transformation. The matrix, and itsclose relative the determinant, are extremely important

concepts in linear algebra, and were first formulated bySylvester (1851) and Cayley.


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In his 1851 paper, Sylvester wrote, "For this purpose wemust commence, not with a square, but with an oblongarrangement of terms consisting, suppose, of linesand columns. This will not in itself represent a determinant, but

is, as it were, a Matrix out of which we may form various systemsof determinants by fixing upon a number , and selecting atwill lines and columns, the squares corresponding of th order."Because Sylvester was interested in the determinant formedfrom the rectangular array of number and not the array itself(Kline 1990, p. 804), Sylvester used the term "matrix" in its

conventional usage to mean "the place from which somethingelse originates" (Katz 1993). Sylvester (1851) subsequently usedthe term matrix informally, stating "Form the rectangular matrixconsisting of rows and columns.... Then all the determinantsthat can be formed by rejecting any one column at pleasure outof this matrix are identically zero." However, it remained up to

Sylvester's collaborator Cayley to use the terminology in itsmodern form in papers of 1855 and 1858 (Katz 1993).


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There are further applications of matrices;

some of these involved the Graph Theory, where the

adjacency matrix of a finite graph is a basic notationof theory. In Physics, there are the linearcombinations of quantum states where the firstmodel of quantum mechanics by Heisenberg in 1925represented the theorys operators by infinite-

dimensional matrices acting on quantum states. Thisis also referred to as matrix mechanics. There is alsomatrix in computer graphics, where 4x4transformation rotation matrices are commonly usedin computer graphics. Other applications of Matrix

are the use of Row reduction, Cramers Rule orDeterminants and using the inverse matrix and lastly,Cryptography.


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In geometry, Heron's (or Hero's) formula,

named after Heron of Alexandria,states that

the area T of a triangle whose sides havelengths a, b, and c is

where s is the semi perimeter of the

triangle:


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The formula is credited to Heron (or Hero) ofAlexandria, and a proof can be found in hisbook, Metrica, written c. A.D. 60. It has been suggested

that Archimedes knew the formula over two centuriesearlier, and since Metrica is a collection of themathematical knowledge available in the ancient world, itis possible that the formula predates the reference givenin that work. Heron's proof (Dunham 1990) is ingenious

but extremely convoluted, bringing together a sequenceof apparently unrelated geometric identities and relyingon the properties of cyclic quadrilaterals and righttriangles. Heron's proof can be found in Proposition 1.8of his work Metrica(ca. 100 BC-100 AD). Thismanuscript had been lost for centuries until a fragmentwas discovered in 1894 and a complete copy in 1896(Dunham 1990, p. 118).


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There are many methods for finding thearea of a triangle and the proponentsconducted an experiment to identify the moreconvenient way of solving the area of thetriangle. The methods suggested for solving thearea are Herons Formula and Matrix.

In Herons Formula, the method startedwith determining the distances between the

given points and the distances are summed upand divided by 2 in order to find the semiperimeter and lastly Herons Formula is used.


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In Herons Formula, the area T of triangle

whose sides are a, b, and c:

and s is the semi perimeter of the triangle:


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In Matrix, the coordinates are arranged

where the xs of all points are in the first row,

the ys in the second row and the lastcolumn are all ones, then evaluate the

determinant and lastly divide it by 2 resulting

to the area of the triangle.In Matrix,


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Both methods are tested according to

the following parameters:

Time

Accuracy

Percentage Error

The proponents presented ten sample

problems for solving the area of the trianglewith different givens of coordinates of a

triangle.


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The conducted experiment claimed to grant thefollowing to the parameters:

To measure the time, proponents introducedproblems to be answered with recorded time resultingin the comparison of time consumed by the twomethods.

To verify accuracy, there are ten problems given. Thesuccessfully answered problems are divided by thenumber of problems and multiplied to 100.

To determine the percentage error, proponents used

the formula of percentage error by subtracting theexact value of the answer from the approximate valueand divided by the exact value and multiplied by 100.


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The following are the recorded data of

parameters using the Matrix Method:

Table 1Parameters Calculated for Matrix

Problem Time(min:secs) PercentageError1

01:05.97

0%

2 00:54.54 0%3 00:48.82 0%4 00:53.39 0%5 00:59.97 0.5%6 00:55.17 0%7 00:47.07 0%8 00:39.96 0%9 00:45.99 0%

10 00:49.00 0%


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The following are the recorded data of parameters using

Herons Formula:

Table 2Parameters Calculated for HeronsFormula

Problem Time(min:secs)

Percentage

Error1 02:59.02 0%2 02:00.73 0%3 01:43.09 0%4 01:24.64 0%5 01:41.29 0%6 01:40.03 0.2%7 01:42.24 0%8 01:08.85 0.2%9 01:42.10 0%

10 01:35.26 0.2%


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The following are the interpretations of data:

In Table 1, the calculated parameters forMatrix showed results in accordance to thetime consumed and percentage error. Amongthe ten problems, problem 5 established a0.5% percentage error.

In Table 2, the parameters calculatedshowed a huge difference to the parametersestablished in Table 1. Problems 6, 8 and 10produced 0.2% percentage error.


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Comparative Results between the Two

Methods in Solving for the Area

MatrixHerons

FormulaAccuracy 90% 80%

Average Time

(min:secs) 00:52.12 01:45.45Percentage Error 0.5% 0.6%


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According to the data table, the resultspresented gave an understanding about thecomparison between methods with regards to the

accuracy, time and percentage error.Based on the results:

In terms of accuracy, the computed percentage is90% in Matrix and 80% in Herons Formula.

In terms of time, the computed time for Matrix is52.12 seconds while 1 minute and 45.45 secondsfor Herons Formula.

In terms of percentage error, the computedpercentage for Matrix is 0.5% while 0.6% forHerons Formula.


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The following conclusions are drawn from following thestudy:

Familiarity in the usage of either two methods shortens thetime consumed in solving for the area of the triangle.

Matrix Method had a higher accuracy than of HeronsFormula.

The time required for solving the area using the Matrix isproved to be less than a minute.

The use of Matrix Method minimized the percentage of error.

Herons Formula had a lower accuracy than of the MatrixMethod.


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Herons Formula consumed more time forsolving the area.

Herons Formula had a larger percentage

error than of the Matrix Method.There is a 10% interval between the two

methods in terms of accuracy.There is 53.03-second interval between

the two methods in terms of time.There is 0.1% interval of percentage of

error between the two methods.


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Based on the analysis of results of the

calculated parameters, it is recommended that:

Matrix Method should be used in finding the areaof the triangle.

Matrix Method should be used to obtain resultsthat are close to the exact value.

Use Matrix Method to save time and effort.

Matrix Method can provide accurate answers.


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[1] En.wikipedia.org.2013. Matrix(mathematics).[online]Available at:http://en.wikipedia.org/wiki/Matrix_(mathematics)[Accessed: 5 Oct 2013].

[2] Heath, Thomas L. (1921). A History of Greek Mathematics(Vol II). Oxford University Press. pp. 321323.

[3] Weisstein, Eric W. "Matrix." From Math World--A WolframWeb Resource. http://mathworld.wolfram.com/Matrix.html

[4] Weisstein, Eric W. "Heron's Formula." From Math World--AWolfram WebResource. http://mathworld.wolfram.com/HeronsFormula.ht

ml

[5] Slideshare.net. 2013. Matrices And Application Of Matrices.[online] Available at:http://www.slideshare.net/mailrenuka/matrices-and-

application-of-matrices [Accessed: 6 Oct 2013].

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