1. Explain why vectors QR and RQ are not equivalent.

Figure 1. Vector                    Figure 2. Vector

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Since a vector is a mathematical object that has magnitude and direction, vectors , as shown in figures 1 and 2, respectively cannot be the same. They may have the same magnitude but their direction is opposite. Since a vector carries both magnitude and direction, the two must be equal in order for them to be equal. Vectors can best be imagined if we put them in “polar form” where a vector is expressed in terms of its angle and magnitude (Mahler & Wesner, 1994). So that where  is the magnitude and  is the angle. On the other hand where  is the magnitude and  is the angle. The negative angle denotes that they are in opposite direction. So clearly

2. Explain in you own words when the elimination method for solving a system of equations is preferable to the substitution method.

“The elimination method consists of eliminating n number of unknowns from n number of equations one by one, going down in upside down staircase method” (CSU Website, 2007). The substitution method requires that you solve one unknown in terms of the other and substitute the result to the next equation in order to eliminate one unknown from two equations. You do the same calculation and substitution with the remaining pairs of equations if the equations are more than two equations.  Therefore as more equations are involved, you do the calculation and substation to as many pairs are available. Then you do it again with the remaining equations for the remaining unknowns, calculate one unknown and substitute which constitute a lot of things to manipulate. Therefore if it gets longer, the chances of committing errors due to the complexity becomes higher. On the other hand, elimination method is more organized. You can immediately generate the multipliers to eliminate one variable at a time. It is less confusing because it is more organized as the number of equations increases. Since the process is basically just finding the multipliers and performing subtraction, the process can easily be automated using computer programs. Elimination also involves smaller steps compared to substitution method. The calculation of one variable in the substitution method is very difficult to automate. So in overall the elimination method is preferred by many over the substitution method.

References

Unkown. (2007). “Vector Methods” . Retrieved from the Engineering Mechanics – College of Engineering & Technology, University of Nebraska-Lincoln website: http://em-ntserver.unl.edu/Math/mathweb/vectors/vectors.html,  on March 19, 2007

Wesner, T. H. & Mahler, P.H. (1994). ”Vectors”. In College Algebra & Trigonometry with Applications. (p. 368). IOWA : Wm. C. Brown Communications, Inc.

Unkown. (2007). “Solving Sytems of Linear Equations” . Retrieved from the Department of Mathematics – California State University website: http://www.math.csusb.edu/math110/src/systems/solving.html,  on March 19, 2007