Programme - HND Electronic Engineering

Module/Unit - Unit 35 - Further Analytical Methods J/601/1465

Level - QCF Level 5

Bob and Carol are attempting to solve a series of simultaneous equations using matrices to represent the equations, and provide a quick method of solution. Neither Bob nor Carol is familiar with this technique. Your task is to help them both by demonstrating how matrices work.

(a) To demonstrate the basic principles of matrices, show Bob and Carol how to:

(i) add together the two matrices: (2  4) + (-2  7)
(ii) Multiply the following matrices: (3  5)   x   (2  -5)
(2 4)        (5  2)

(iii) Find the determinant of the following matrix (5  9)
(4  4)
(iv) Find the inverse of the following matrix (3  2)
(-2 -8)

(b) Bob has two simultaneous equations which need to be solved. By re-arranging them and forming a matrix equation, show him how you could use matrix algebra to solve them.

3x - 5y = -17.6
-2x + 7y - 22 = 0

(c) Carol has three simultaneous equations to solve and wants to use a technique called Gaussian Elimination" to do this. The three equations are:

3x + 2y - 3z = -7
x + 4 y + z =25
2x + 6y + 3z =49

Show her how to use Gaussian elimination to solve these equations.

(d) For (b) and (c), use methods with which you are familiar to check your solutions to both parts. Further to this, critically reflect on the advantages and disadvantages of the techniques used to solve these problems. What are the main advantages and disadvantages to these analytical methods and what alternatives might you suggest?

Opportunities for Higher Gradinq

D1 In part (d) you need to critically reflect on your work, principally to your solutions to (b) and (c). A formal analysis is required here, suggesting alternative approaches with which you are familiar.

Ted and Alice are working on some problems which involve representing systems of forces represented by vectors.

Your task is to help them solve the following problem:

(a) LO3.3

A force of (3i + 2j) Newtons acts at the point with position vector (2i - j) metres and a secon force of (-3i + j) Newtons acts at the point with position vector (-3i - j) metres. Find the moment of the system about the point with position vector (I + 2j).

Alice is working on a new design of roof truss. She explains to Ted how the forces acting in the wooden sections could be resolved into vertical and horizontal components using trigonometry. Use this technique to answer the question below.

(b) LO3.1 Find the magnitude and direction of the resultant of the set of forces shown below

CHECKLIST

(a) Basic manipulation of matrices completed without error
(b) Solution of simultaneous equations completed successfully using a matrix method
(c) Solution of three simultaneous equations correctly using Gaussian elimination
(d) Independently find appropriate methods of checking answers to (h) and (c) above indicating strengths and weaknesses in approaches

(a) Calculations carried out successfully using vector approach and overall moment of system determined correctly
(b) magnitude and direction of the resultant of the set of forces calculated correctly

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#### Are You Looking for Unit 35 - Further Analytical Methods - J/601/1465, QCF Level 5 ?

The problems provided in the questionnaire are solved and the solutions provided as below:

Observations:

Solution#1

A) While calculating the total moment of two system of forces, the moment of each force about  the point O calculated and later the moments are added to find the total moment due to the two forces. For calculating the moments vector product of force and the distances are used and totaled.

While calculating the resultant of the system of forces, each of the forces present there in the system is resolved along the X and Y direction and finally the resultant of all the three forces is calculated.

Resultant force magnitude and the angle it makes with the X-axis in anticlock wise direction(Negative ) calculated and indicated there in the solution.

Solution#2

(i) Matrix addition, multiplication, inverse calculations are directly performed.

(ii) Linear equations of two unknowns and three unknowns are also calculated directly from the problem conditions.

(iii) Also the solutions found are verified from the traditional analytical procedures.

A) F1 = (3i +2J) A = (2i - j) F2=(-3i +J) B = (-3i -J)

The given point for moment calculation = (I + 2J)

Moment = OA X F1 + OB X F2

= (i-3J) X(3i +2J) + (-4i -3J) X (-3i + J)

= 11K + 5K

=16K

B) Determination of the resultant of the Forces:

From F.B.D shown, the

Resolving all the forces along X and Y directions,

Rx = 4 cos30 + 3 Cos10 - 5 Sin20

Ry = 4 Sin30 - 3 Sin10 - 5 Cos20

Rx = 4.71N

Ry = -3.22

Hence the magnitude of R = |R| = sqrt(Rx^2 + Ry^2) = 5.71

Tan(theta) = Ry/Rx

Theta = -34.350

Angle is negative implies in anti-clock wise direction.

A)  (1) Matrix Addition

(2 4) + (-2 7) = (0 11)

(2) Matrix Mulitplication

(3  5)  (2  -5)  =  (6+25   -15+10) = (31  -5)
(2 4)   (5  2)       (4+20  -10 + 8)     (24 - 2)

(3) Determinant of the matrix

|5  9| = 5*4 - 9 *4 = -16
|4  4|

(4) Inverse of  (3  2) =  [ 1/(3*8 -2 *((-2)) ] * (8 -2)= (1/28)* = (8 -2)
(-2 -8)                                        (2 3)                     (2 3)

(2/7   -1/14)
=  (1/14    3/28)

(B) Given Equations

3x -5y =-17.6

-2x +7y =22.0

Matrix form:

(3   -5)    (x)        (-17    .6)
(-2   7) *  (y) =    (22     .0)

Hence by Rearranging the terms,

A*X = B

X = Inv(A) * B

Inv (A) = [1/(21-10)]* (7   5) =   (1/11   5/11)
(2  3)          (2/11    3/11)

X = Inv (A) *B

X = 7/11*(-17.6) + 5/11* 22.0 = -1.2

Y = 2/11*(-17.6) + 3/11* 22.0 = 2.8

(C)  Given Equations

3x + 2y -3z =-7  -------(R1)

X +4y +z = 25 ----------(R2)

2x +6y +3z =49 ---------(R3)

Making the following set of operations,

R1--------R1/3

R2--------R2 - R1/3

R3 ---------R3-2R1

R2----------3/10 R2

R3--------------R3 -14/3 R2

Hence,  it yields,

Which further on simplification will yield,

Z= 7, y = 4 and x =2

Hence the solution matrix

(2)
X =  (4)
(7)

(D )There do exist alternative procedures for solving the equations set given in the set B and set C, the following procedures indicate how to solve the equations set,

3x -5y = -17.6

-2x +7y =22.0

Multiply the first equation by 2 and second equation by 3 and adding the two will provide an equation,

-10y +21y = -17.6 x2 +22 x3 = 30.8

Or alternatively +11y =30.8

Y =2.8

Alternatively on substitution in equation 1 it will provide 3x = -17.6 + 5*2.8  =-3.6

Which indicate that x = -1.2, hence the result is correct as per matrix methods.

3x + 2y -3z =-7

X + 4y +z =25

Multipling the second equation by 3 and subtracting the first will give

-10y + 6Z =82-----------------(1)

Also

X + 4y +z =25

2x + 6Y +3Z =49

Multiplying the second by 1 and from it subtracting the twice of first equation will provide

-2y + z = -1 -------------------- (2)

From 1 and 2 equations,

X = 2 and  y =4 and z=7, which indicates that the matrix formulation is correct and the procedures employed before are correct.

Analysis of the procedures employed: