Engineering Mathematics 2 Basic Concepts PDF

Engineering Mathematics 2 Basic Concepts MPS Department | FEU Institute of Technology  OBJECTIVES After the completion of the topics the student can: Plot / locate points on the cartesian coordinate plane Compute for the distance between two points L...

Engineering Mathematics 2 Basic Concepts MPS Department | FEU Institute of Technology  OBJECTIVES After the completion of the topics the student can: Plot / locate points on the cartesian coordinate plane Compute for the distance between two points Locate points that divides a line segment Compute for the area of a polygon given coordinates A coordinate system that specifies each point uniquely in a plane by a set of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in the same unit of length. Each reference line is called a coordinate axis or just axis (plural axes) of the system, and the point where they meet is its origin, at ordered pair (0, 0). The coordinates can also be defined as the positions of the perpendicular projections of the point onto the two axes, expressed as signed distances from the origin. The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra. Using the Cartesian coordinate system, geometric shapes (such as curves) can be described by Cartesian equations: algebraic equations involving the coordinates of the points lying on the shape. For example, a circle of radius 2, centered at the origin of the plane, may be described as the set of all points whose coordinates x and y satisfy the equation x2 + y2 = 4. Cartesian coordinates are the foundation of analytic geometry, and provide enlightening geometric interpretations for many other branches of mathematics, such as linear algebra, complex analysis, differential geometry, multivariate calculus, group theory and more. A familiar example is the concept of the graph of a function. Cartesian coordinates are also essential tools for most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more. They are the most common coordinate system used in computer graphics, computer-aided geometric design and other geometry-related data processing. The adjective Cartesian refers to the French mathematician and philosopher René Descartes, who published this idea in 1637. It was independently discovered by Pierre de Fermat, who also worked in three dimensions, although Fermat did not publish the discovery.The French cleric Nicole Oresme used constructions similar to Cartesian coordinates well before the time of Descartes and Fermat. Both Descartes and Fermat used a single axis in their treatments and have a variable length measured in reference to this axis. The concept of using a pair of axes was introduced later, after Descartes' La Géométrie was translated into Latin in 1649 by Frans van Schooten and his students. These commentators introduced several concepts while trying to clarify the ideas contained in Descartes' work. The development of the Cartesian coordinate system would play a fundamental role in the development of the calculus by Isaac Newton and Gottfried Wilhelm Leibniz. The two- coordinate description of the plane was later generalized into the concept of vector spaces. The axes of a two-dimensional Cartesian system divide the plane into four infinite regions, called quadrants, each bounded by two half-axes. These are often numbered from 1st to 4th and denoted by Roman numerals: I (where the signs of the two coordinates are I (+,+), II (−,+), III (−,−), and IV (+,−). When the axes are drawn according to the mathematical custom, the numbering goes counter- clockwise starting from the upper right ("north-east") quadrant. Distance directed 𝑷 𝑷 𝑃 1𝑃 = −𝑃1𝑃2 2 𝑃1𝑃2 = 𝑥2 − 𝑥1  𝑃1𝑃2 = 𝑥2 − 𝑥1 Find the directed distance from A to B. 1. 𝐴 6 & 𝐵 (−4) 2. 𝐴 − 48 & 𝐵 − 12 3. 𝐴 𝑘 & 𝐵 (3𝑘) 4. 𝐴 𝑥 + 𝑦 &𝐵 𝑥 + 𝑦 5. 63 & 𝐵 − 28  𝑃 1 𝑃 2 = 𝑥 2 − 𝑥1 Find the undirected distance from A to B. A B C D E F -5 -1 0 3 8 12 Find: AD, CA, FC, CE, BF A square of side 2 units have one vertex a (0,0) and one diagonal along the x-axis. What are the coordinates of the three (3) vertices? Answer: 2, 2 2 ,− 2 2 2 ,0 Case 1: 𝑃1 𝑋, 𝑌1 & 𝑃2 𝑋, 𝑌2 𝑑 = 𝑌2 − 𝑌1 →Therefore, it is a VERTICAL LINE Case 2: 𝑃1 𝑋1, 𝑌 &𝑃2 𝑋 2, 𝑌 𝑑 = 𝑋 2 − 𝑋 1 →Therefore, the line is HORIZONTAL LINE Case 3: When neither the X’s nor Y’s are the same. 𝑃1 𝑋1, 𝑌1 & 𝑃2 𝑋2, 𝑌2 𝑑= 𝑋 2 − 𝑋 1 2 − 𝑌2 − 𝑌1 2 →Therefore, the line is OBLIQUE Solve for the ff, and classify what type of case line. 1. −9,6 3,9 2. 2,13 4,5 3. 0,0 9,18 4. −6,8 14,8 5. 0,12 0,28 1. Show if the ff. points are the vertices of a right triangle on A (4,-3) , B (0,0) and C (3,4). 2. Compute the distance: D (-2,0), E (1,-2), and F (5,4) 3. Show the ff. points: A (-1,2), B ( 3,0), and C (7,-2) are collinear points. 4. Show if the A (0,1), B (2,5), and C (-1,4) are vertices of an isosceles triangle. 5. Show that the Quadrilateral whose vertices are A (-2,6), B (4,3), C (1,-3), and D (-5,0) is a square 6. Find the point on the y-axis that is equidistantfrom A (6,1), and B (-2,3).  𝑥 − 𝑥1 : 𝑥 − 𝑥 2 =𝑚: 𝑛 External Division Point Therefore −𝑛𝑥 1 +𝑚𝑥 2 𝑥= 𝑚−𝑛 −𝑛𝑦 1 +𝑚𝑦 2 y= 𝑚−𝑛 Point of division: is a point that divides a line segment to a given ratio. From similar triangles, we can find the x-coordinate of P as follows. 𝑃1𝑃 𝑥 − 𝑥1 = 𝑃1𝑃2 𝑥2 − 𝑥1 𝑥−𝑥 1 Let 𝑟 = 𝑥 2 −𝑥 1 𝑥 = 𝑥1 + 𝑟(𝑥2 −𝑥 1 )  𝑥 − 𝑥1 : 𝑥 − 𝑥 2 =𝑚: 𝑛 External Division Point Therefore −𝑛𝑥 1 +𝑚𝑥 2 𝑥= 𝑚−𝑛 −𝑛𝑦 1 +𝑚𝑦 2 y= 𝑚−𝑛  𝑥 − 𝑥1 : 𝑥 2 − 𝑥 =𝑚: 𝑛 Internal Division Point Therefore 𝑛𝑥 1 +𝑚 𝑥 2 𝑥= 𝑚+𝑛 𝑛𝑦 1 +𝑚 𝑦 2 y= 𝑚+𝑛 Recall…. 1. 3AB = 5AC 2. 3AB = 5AC 2 4 3. 2AC = 3BA 4. AB = 3BC 3 5. AC = CB 5 BA 3 6 BC = 2. 1. Find the point B on AC such that the ratio of AB to BC is 3:1 where A (9,5) and C (-7,1) 1 2. Find the point B on AC such that the ratio of AB = AC 5 where C (-7,-3) and A (8, -8) 3. Find the (external )point C such that the ratio of AB to BC is 1:1 where A(1,-9) and B(2,0). 4. Points A, B, C are collinear points. B is in between AC. The ratio of AB:AC is 1:4 where A(-7,-8) and B(-3,-5) 5. Find B on line segment AC such that 6. AB:BC =3:1 where A(-7,2) and C(3,-2) 7. Find the coordinates of Point B on line segment AC such 1 that AB = AC where A(8,1) and C(-7,7) 3 Base and height are perpendicular. Area of Polygon 1 𝑥 1 𝑦 1 1 𝑥 1 𝑦 1 1 𝑨= 𝑥2𝑦21 𝑥2𝑦21 2 𝑥3𝑦31 𝑥3𝑦31 1. What is the area of the triangle with vertices (1,2) (-2,1) (3,0) 2. What is the area of the polygon with vertices (3,0)(2,3) (-2,-1) 3. F 2,0 , E 6,4 , U (3,2)  1 𝑥1 𝑥2 𝑥2 𝑥3 𝑥𝑛 𝑥1 𝐴 =± 𝑦1 𝑦2 + 𝑦2 𝑦3 + ⋯+ 𝑦𝑛 𝑦1 2 𝐸𝑞. 1: 𝑥1𝑦2 + 𝑥 2𝑦 3 + ⋯+ 𝑥 𝑛𝑦 1 𝐸𝑞. 2: 𝑦1𝑥2 + 𝑦2𝑥3 + ⋯+ 𝑦 𝑛𝑥1 1 𝐴 =± 𝐸𝑞. 1 − 𝐸𝑞. 2 2 1. A (3,5) B (6,8) C (9,6) D (5,0) 2. T (0,8) U (6,5) V (6,-1) W (-2,-4) X (-5,3) 3. C (-3,10) D (9,4) E (9,-10) 4. R (-4,4) S (-9,-1) T (-4,-6) U(1,-1) Brannan, David A.; Esplen, Matthew F.; Gray, Jeremy J. (1998), Geometry, Cambridge: Cambridge University Press, ISBN 978-0-521- 59787-6 Burton, David M. (2011), The History of Mathematics/An Introduction (7th ed.), New York: McGraw-Hill, ISBN 978-0-07-338315-6 Smart, James R. (1998), Modern Geometries (5th ed.), Pacific Grove: Brooks/Cole, ISBN 978-0-534-35188-5 Descartes, René (2001). Discourse on Method, Optics, Geometry, and Meteorology. Translated by Paul J. Oscamp (Revised ed.). Indianapolis, IN: Hackett Publishing. ISBN 978-0-87220-567-3. OCLC 488633510. Korn GA, Korn TM (1961). Mathematical Handbook for Scientists and Engineers (1st ed.). New York: McGraw-Hill. pp. 55– 79. LCCN 59-14456. OCLC 19959906. Margenau H, Murphy GM (1956). The Mathematics of Physics and Chemistry. New York: D. van Nostrand. LCCN 55-10911. Moon P, Spencer DE (1988). "Rectangular Coordinates (x, y, z)". Field Theory Handbook, Including Coordinate Systems, Differential Equations, and Their Solutions (corrected 2nd, 3rd print ed.). New York: Springer-Verlag. pp. 9–11 (Table 1.01). ISBN 978-0-387- 18430-2. Morse PM, Feshbach H (1953). Methods of Theoretical Physics, Part I. New York: McGraw-Hill. ISBN 978-0-07-043316-8.LCCN 52- 11515. Sauer R, Szabó I (1967). Mathematische Hilfsmittel des Ingenieurs. New York: Springer Verlag. LCCN 67-25285. https://en.wikipedia.org/wiki/Cartesian_coordinate_system#History