3D Vectors: A Simple Mathematical Investigation Assessment Answer

Introduction :

The mathematical elements consisting of Magnitude as well as direction are termed as Vectors. We can use vectors in 3 dimensional space in many ways like creating real – life events created by dynamic models using vectors. For this investigation report, we are going to use vector analysis for representing the velocity and position of 2 mosquitoes which are Mosquito – A and Mosquito – B. We are going to determine there distances at any particular time (t) and the distance they are apart from each other. Since, the path of the mosquitoes are two straight lines that are skew lines as well, therefore, closed distance of approach will be determined, thereafter, repellent represented by another straight line is going to be implemented at such an optimal position in order to affect both the mosquitoes with maximum capacity possible. Another method (Conical approach) will be explored in place of line model in order to determine precise positions and angle of spray needed at the moment. Finally, differential calculus will be used to check the effects of gravitational force on the paths of the repellant as well as the mosquitoes will be considered. All the methods used in this investigation will be 3 dimensionally represented by GeoGebra. 

The main aim of the investigation is to investigate the direction of spray to be used both via plane movement and via line movement. In order to conduct the mathematical investigation, techniques such as Vector analysis, line equations in 3 dimensions, modulus of a vector, parametric equations as well as 3D grapher such as GeoGebra is going to be used. All the analysis, results, mathematical calculations, interpretation and the using of graphing software is going to be investigated through this report.

For many years, it has been hypothesized that mosquito repellant is very harmful to animals, insects as well as humans too. The most popular chemical used in a mosquito repellant is : n, n dimethyl meta toluamide (DEET). The chemical is most efficient in killing bugs, on the other hand, it can cause skin and eye irritation to humans as well as soon as it gets in contact with scrapes and cuts. It has been scientifically proved as well that the chemical can interfere in the body enzymes and can cause changes in the human nervous system hence impacting the human life for long time giving rise to Chronic diseases as well. The most popular brand that is in use is Picaridin. The spray repellant has its effect from 2 to 8 hours and hence can cause severe damage to the humans as well as pet animals. Hence, it is the responsibility of all of us to use the spray very much efficiently so that the purpose of killing the mosquito is fulfilled without causing any severe damage to humans. 

PART 1

For some (sensibly) chosen values of find the parametric equations for the position of each mosquito at time t, where t is the time in seconds, using centimetres for the distance.

What is the speed of each mosquito and how long does it take each mosquito to reach a person (or two people) at a point of your choice along its straight line path.

Assuming that the person-1 is sitting at the location : (13, 16, -6) and person-2 is sitting at the location : (40, 49, -6). We have-

For Mosquito - A :

Initial Position Vector : sA = (2t + 5, 4t, -3t + 6)

Velocity vector : vA = (2, 4, -3)

Speed of Mosquito – A : |vA| = 22+ 42+ (-3)2 = 5.385 cm/s

At t = 0, location of Mosquito – A : sA|t=o = (5, 0, 6)

Distance vector between person and mosquito = ((13 – 5), (16 – 0), (-5 – 6)) = (8, 16, -11)

Distance between person-1 and mosquito-A = 82+ 162+ (-11)2 = 21 cm

Time taken by Mosquito-A to reach person-1(t1) = 215.385 = 3.900 sec

For Mosquito – B :

Initial Position vector : sB = (5t + 5, 7t, -t + 1)

Velocity vector : vB = (5, 7, -1)

Speed of Mosquito – B : |vB| = 52+ 72+ (-1)2 = 8.660 cm/s

At t = 0, location of Mosquito – B : sB|t=o = (5, 0, 1)

Distance vector between person and mosquito = ((40 – 5), (49 – 0), (-6 – 1)) = (35, 49, -7)

Distance between person-2 and mosquito-B = 352+ 492+ (-7)2 = 60.622 cm

Time taken by Mosquito-B to reach person-2(t2) = 60.6228.660 = 7.000 sec

PART 2

If at time t, mosquito A is at position P and mosquito B is at position Q, determine the vector and find their distance apart at any instant. Find the time when the mosquitoes are closest to each other and their distance apart at this instant.

At time = t,

Position Vector of Mosquito-A : sA = (2t + 5, 4t, -3t + 6)

Position Vector of Mosquito-B : sB = (5t + 5, 7t, -t + 1)

Distance vector from mosquito A to B is calculated as-

DAB = (((5t + 5) - (2t + 5)), (7t – 4t), ((-t + 1) – (-3t + 6)))

= (3t, 3t, (3t – 5))

Calculating the separation distance magnitude at time = t, we get-

|DAB| = (3t)2+ (3t)2+ (3t-5)2

|DAB| = 27t2-30t+ 25 …(1)

Equation-(1) represents the separation distance magnitude between the two mosquitoes at time = t.

Equation-(1) can be re-written as-

DAB2 = 27t2 – 30t + 25

Calculating maxima/ minima for DAB is equivalent of calculating it for DAB2.

Let f(t) = DAB2 = 27t2 – 30t + 25

For calculating critical points, we have-

dfdt = 54t – 30 = 0

t = 3054 = 0.556 s

Also, d2fdt2|t=0.556 = +ve (Point of Minima)

Also, |DAB|(t = 0.556) = 27(0.556)2 – 30(0.556) + 25 

= 16.667 cm

Therefore, both the Mosquitoes are closest at t = 0.556 s and the closest distance between the two mosquitoes comes out to be 16.667 cm. 

PART 3

Now investigate the best location for you to be holding the insect repellant to hit both mosquitoes with just one spray. The sprayed repellent may be represented by a line in 3D. Designate a position and velocity of sprayed insect repellant so that both mosquitoes will be within its range.

Modify your model if possible to consider the insect spray as a plane in 3D or a 3D shape.

Spray must be placed at midpoint of the two mosquitoes when they are closest.

Position vector of Mosquito-A at t = 0.556s : sA(t = 0.556) = (6.112, 2.224, 7.668)

Position vector of Mosquito-B at t = 0.556 : sB(t = 0.556) = (7.780, 3.892, 0.444)

Position of spray = Midpoint of above two locations

= (6.112 +7.780 2,2.224 + 3.8922,7.668 + 0.4442)

= (6.946, 3.058, 4.056)

Velocity vector = (3.5, 5.5, -2)

3D diagram of the Propellant is shown below in Figure-1 :

Figure-1

Now, considering 3D plane for the motion of the repellant vector can be calculated as :

Finding two vector in the plane from the repellant point towards the direction of the mosquitoes. Let those two vectors be P and Q. Then,

P = sA – position vector of repellant

= (6.112 – 6.956, 2.224 – 3.058, 7.668 – 4.056)

= (-0.844, -0.834, 3.612)

Q = sB – position vector of repellant

= (7.780 – 6.956, 3.892 – 3.058, 0.444 – 4.056)

= (0.824, 0.834, -3.612)

Taking there cross product to find the normal vector of the plane equation-

P x Q = i j k -0.844 -0.834 3.612 0.824 0.834 -3.612

= (0)i – (0.072)j + (-0.017)k

= -0.072j – 0.017k

Using equation of plane standard form-

ax + by + cz = d

d = (0)(6.956) + (-0.072)(3.058) + (-0.017)(4.056)

= -0.289

Therefore, we get the equation of plane of the repellant as-

0.072x + 0.017y = 0.289

72x + 17y = 289 is the required equation.

PART 4

Analyse and interpret your results, including consideration of the reasonableness and limitations of the results.

The position vector and the velocity vector for both the mosquitoes were given. There paths were analyzed and the location of closest approach was determined using calculus and vector algebra. Finally, a mosquito repellant was placed at the same location in order to bring both the mosquitoes in the range of the spray of propellant. Finally, the direction of the velocity of the spray was determined using the average velocity vectors of both the mosquitoes. Hence, using vector algebra, we are able to move the resources efficiently.

There were several assumptions that we made to carry out our mathematical investigation. The first one is, we assumed that mosquitoes are travelling in a straight line but in real life scenario, there path is random and we can’t predict it. Hence, we can use statistical method of random motion or the theory of Bezier curves in order to determine the path of the mosquitoes and then apply vector analysis, which would be beyond the theory of this course. The second assumption, we made is that, the room size is infinitely large, hence, the straight lines of motion are never ending lines. Hence, the path of the mosquitoes were not defined by the dimensions of the room which implies that they could fly anywhere. So, we didn’t consider collision of mosquitoes with walls or other obstacles in the room. On considering real life situation considering the size of the room, we need to include the limits around the function as well. Alternatively, we can assume that mosquitoes are flying in an open area above the ground but still, the ground boundary must be considered. Third assumption we made is, there is no drag force acting on the mosquitoes but if we consider it practically, the wind drag force have significant impact on the path of the mosquitoes. 

The method we modelled have effectively considered the motion and shape of objects in 3 - dimensional space as shown in Figure-2 and Figure-3, considering all the linear paths of mosquitoes as well as the repellant. In case we consider the gravitational force, the path of a projectile or the motion of mosquitoes could be parabolic paths. But since, the mass of the mosquitoes is very less which results to insignificant gravitational force acting on the mosquitoes as well as the repellant droplets, hence, those parabolic paths becomes straight lines since we observe a very small part of that parabolic curve. Therefore, we can neglect the impact of the gravitational force acting on them. In the whole investigation, we have used reasonable distances between the mosquitoes, humans and repellants which is in the range of centimeters to few meters.

Figure-2

Figure-3

Complete a report for the mathematical investigation. 

The investigation report should be a maximum of 15 single-sided A4 pages if written, or the equivalent in multimodal form.

The report may take a variety of forms, but would usually include the following:

• an outline of the problem and context
• the method required to find a solution, in terms of the mathematical model or strategy used
• the application of the mathematical model or strategy, including

• relevant data and/or information
• mathematical calculations and results, using appropriate representations
• the analysis and interpretation of results, including consideration of the reasonableness and limitations of the results

• the results and conclusions in the context of the problem.

A bibliography and appendices, as appropriate, may be used.

The format of an investigation report may be written or multimodal.

The report, excluding bibliography and appendices if used, must be a maximum of 15 A4 pages if written, or the equivalent in multimodal form. The maximum page limit is for single-sided A4 pages with minimum font size 10. Page reduction, such as 2 A4 pages reduced to fit on 1 A4 page, is not acceptable. Conclusions, interpretations and/or arguments that are required for the assessment must be presented in the report, and not in an appendix. Appendices are used only to support the report, and do not form part of the assessment decision.

Assessment Design Criteria

Concepts and Techniques

The specific features are as follows:

CT1 Knowledge and understanding of concepts and relationships.

CT2 Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts.

CT3 Application of mathematical models.

Reasoning and Communication

The specific features are as follows:

RC1 Interpretation of mathematical results.

RC2 Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations.

RC3 Use of appropriate mathematical notation, representations, and terminology.

RC4 Communication of mathematical ideas and reasoning to develop logical arguments.

Performance Standards for Stage 2 Specialist Mathematics

-	Concepts and Techniques	Reasoning and Communication
A	Comprehensive knowledge and understanding of concepts and relationships. Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts. Successful development and application of mathematical models to find concise and accurate solutions. Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems.	Comprehensive interpretation of mathematical results in the context of the problem. Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations. Proficient and accurate use of appropriate mathematical notation, representations, and terminology. Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments. Effective development and testing of valid conjectures, with proof.
B	Some depth of knowledge and understanding of concepts and relationships. Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts. Some development and successful application of mathematical models to find mostly accurate solutions. Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems.	Mostly appropriate interpretation of mathematical results in the context of the problem. Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations. Mostly accurate use of appropriate mathematical notation, representations, and terminology. Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly effective development and testing of valid conjectures, with substantial attempt at proof.
C	Generally competent knowledge and understanding of concepts and relationships. Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts. Successful application of mathematical models to find generally accurate solutions. Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems.	Generally appropriate interpretation of mathematical results in the context of the problem. Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations. Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy. Generally effective communication of mathematical ideas and reasoning to develop some logical arguments. Development and testing of generally valid conjectures, with some attempt at proof.
D	Basic knowledge and some understanding of concepts and relationships. Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts. Some application of mathematical models to find some accurate or partially accurate solutions. Some appropriate use of electronic technology to find some accurate solutions to routine problems.	Some interpretation of mathematical results. Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations. Some appropriate use of mathematical notation, representations, and terminology, with some accuracy. Some communication of mathematical ideas, with attempted reasoning and/or arguments. Attempted development or testing of a reasonable conjecture.
E	Limited knowledge or understanding of concepts and relationships. Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems. Attempted application of mathematical models, with limited accuracy. Attempted use of electronic technology, with limited accuracy in solving routine problems.	Limited interpretation of mathematical results. Limited understanding of the meaning of mathematical results, their reasonableness, or limitations. Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy. Attempted communication of mathematical ideas, with limited reasoning. Limited attempt to develop or test a conjecture.

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