VECTOR GEOMETRY AND MATRIX METHOD Essay Example

VECTOR GEOMETRY AND MATRIX METHOD

Vector Geometry and Matrix Method

Introduction

Engineering situations employ algebra methods to analyse and model variables in various fields such as electrical and mechanical force system, amongst many others. The paper seeks to report on how the idea of vector geometry and matrix as one of the algebraic approaches upon which modern engineering mathematics relies to analyse and model certain situations.

Method

Certain engineering situations require analysis and modelling that would call for mathematical arithmetic as well as the concept of complex numbers. The aspect of modelling the situations would require geometric approaches. In this case, analysis has be done to force with respect to a force
newton acting on a line passing through a point
The analysis intends to determine the moment M and magnitude
of the force about a point
using vector geometry analysis.

The idea of vector geometry and matrix has also been used to analyse data involving alternating voltage; velocity vector of a rotating sphere, and the magnitude and direction of a different velocities with impacts against each other. In this perspective, the matrix and vector approach to complex numbers have been employed. For Task two, the concept of matrix and vector analysis has also been used to provide solution to different tension forces acting in a simple framework. To determine the magnitude of tension forces, Gaussian elimination has been employed.

Calculation and Results

Task ONE

1. For a force
   newton acting on a line passing through a point
   , the analysis to determine the moment M and magnitude
   of the force about a point
   using vector geometry analysis would be as shown below.

Changing the force to
newton would result in a moment and magnitude as demonstrated below.

Graphical representation of the above force acting on a line about a point P and Q, hence impacting moment effect of certain magnitude solved above would be as follows.

2. For the instantaneous values of two alternating voltage given by
   Volts, and
   Volts, their plot in the same axes within a scale of
   Volts and
   would be as follows.

In the same perspective, to obtain a sinusoidal expression that would accommodate
in the form
would be:

3. For a sphere rotating with an angular velocity ꙍ about the z-axis of a system and coinciding at the axis, the velocity vector and its magnitude at a point
   mitres from the axis can be obtained as follows, given that its angular velocity at the position is
   .

4. The magnitude and direction of
   correct to 2 decimal places for
   at 52* and
   at -28* using the concept of complex numbers would be as follows.

Representing the two velocities on a graph produces the positions shown below.

The tensions, and
in a simple framework given by the equations:
;
, and
can be analysed for the magnitude of the tensions using Gaussian elimination as shown below.

(2) Apply – 5 (1)

(2) Apply – 4 (3)

Equation (3) x 5 – Equation (1) x 6

Therefore:

Substituting value into equation 1

Therefore:

Now substituting both value into equation 2

Therefore:

From the elimination method above,

Discussion

About employing vector geometry to analyse and determine moments and magnitude of forces acting about a point in the same line, the idea is to consider a body moving a due to a force acting on it. Due to the force, the body is likely to move in any direction, or it can turn. The turning effect of the force, also referred to as moment is determined about a point, in which a vector geometry analysis enables the view of the point based on direction and distance from the point of force. Therefore, the moment of a line vector, which is the force, about a point P as view from a point Q in the same line of force can be given as vector PQ x F = 0, and can be represented as shown below.

Also, when determining the moment, it should be noted that the choice of a point along the line of action of the force does not affect the results (Bird 2013), but change in force or vector does just as demonstrated in the calculations in the previous section.

Solving for alternating voltages and other variables using sinusoidal expression approach entails the idea of homogenous solutions as well as particular solutions of change in voltage, which is one of the common approaches in electrical systems. Whilst the objective in this aspect is to determine the change in volts in a given electrical system, angular velocity of a rotating body defines the rate of change of angular displacement ꙍ with time (James 2007). Vector also helps determine the direction of action of force, as well as the magnitude, as the sphere, as in this case, rotates about its axis. The idea of coinciding with the x-axis only provides the reference point of viewing vector.

About the use of Gaussian elimination to solve for different tension forces acting in a system, the task is to comprehend how matrix reduction approach can be used to solve linear equations (Kreyszig & Norminton 2011). The method employs certain steps that should be followed towards determining the value of different forces that are represented by given linear equations. In this case, the three equations were solved using matrix.

Conclusion

Engineering situations employ algebra methods to analyse and model variables in various fields such as electrical and mechanical force system, amongst many others. The idea of vector geometry and matrix is one of the algebraic approaches upon which modern engineering mathematics relies as demonstrated in the sections above.

Reference List

Bird, J 2013, ‘Higher engineering mathematics’, Oxford: Routledge

James, G 2007, ‘Modern engineering mathematics’, Pearson Education.

Kreyszig, E & Norminton E 2011, ‘Advanced engineering mathematics (9^th Ed.)’, New York: John Wiley & Sons, Inc.