EM3 Mathematics for Engineers 2 PDF

Pamantasan ng Lungsod ng Valenzuela College of Engineering and Information Technology Electrical Engineering Department EM3: Mathematics for Engineers 2 UNIT 1: Introduction to Analytic...

Pamantasan ng Lungsod ng Valenzuela College of Engineering and Information Technology Electrical Engineering Department EM3: Mathematics for Engineers 2 UNIT 1: Introduction to Analytic Geometry Analytic Geometry – is a branch of Mathematics that combines the principles of Algebra and Geometry. It involves the study of geometric shapes and figures using coordinate system. Rene Descartes – founder of Analytic Geometry by introducing coordinates system in 1637. Rectangular Coordinates System – also known as Cartesian Coordinates System. In two-dimensional Cartesian Coordinate System, each point is represented by an ordered pair of numbers (x, y) wherein x is the horizontal coordinate (abscissa), and y is the vertical coordinate (ordinate). Distance between Two Points in a Plane Using Pythagorean Theorem, the distance between two points can be calculated using: 𝑑2 = (𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 Therefore, 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 Distance Formula Distance between Two Points in Space 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2 + (𝑧2 − 𝑧1 )2 Prepared by: Engr. Cabral, Engr. Celestial & Engr. Cruz Distance between a Line and a Point Consider a point with coordinates (x, y) and a line with equation Ax + By + C = 0. 𝐴𝑥1 + 𝐵𝑦1 + 𝐶 𝑑= ±√𝐴2 + 𝐵2 Use: + : if B is positive and the point is above the line or to the right of the line + : if B is negative and the point is below the line or to the left of the line – : if otherwise Distance between Two Parallel Lines The (perpendicular) distance, d, between the two lines is: 𝐶1 − 𝐶2 𝑑= ±√𝐴2 + 𝐵2 Use the sign (either + or –) that would make the distance positive. Division of a Line Segment Consider two points with coordinates (x1, y1) and (x2, y2). The line segment formed by these two points is divided by a point P whose coordinates are (x, y). Let r1 and r2 be the corresponding ratio of its length to the total distance between two points. The abscissa of the point, P is: (𝑥1 𝑟2 ) + (𝑥2 𝑟1 ) 𝑥= 𝑟1 + 𝑟2 The ordinate of the point, P is: (𝑦1 𝑟2 ) + (𝑦2 𝑟1 ) 𝑦= 𝑟1 + 𝑟2 Midpoint of a Line Segment If the point, P is at the midpoint of the line segment, then the abscissa and ordinate of the point are the following: 𝑥1 + 𝑥2 𝑦1 + 𝑦2 𝑥= 𝑎𝑛𝑑 𝑦= 2 2 Prepared by: Engr. Cabral, Engr. Celestial & Engr. Cruz Slope of a line (m) The slope of a line is defined as the rise (vertical) per run (horizontal). 𝑟𝑖𝑠𝑒 ∆𝑦 𝑠𝑙𝑜𝑝𝑒 = 𝑚 = = 𝑟𝑢𝑛 ∆𝑥 where: ∆ denotes an increment. 𝑦2 − 𝑦1 𝑡𝑎𝑛𝜃 = 𝑥2 − 𝑥1 Note: 1. A line parallel to the x-axis has a slope of zero. 2. A line parallel to the y-axis has a slope of infinity (∞). 3. For parallel lines with slopes of m1 and m2 respectively, the slopes are the same. 𝑚1 = 𝑚2 4. For perpendicular lines with slopes of m1 and m2 respectively, the slope of one is the negative reciprocal of the other. 1 𝑚2 = − 𝑚1 Angles formed by Two Lines Consider two lines with slopes m1 and m2. The angle between these lines (Line 1 and Line 2) may be calculated using the following formula: 𝑚2 − 𝑚1 𝑡𝑎𝑛𝜃 = 1 + 𝑚2 𝑚1 𝑜𝑟 𝑚2 − 𝑚1 𝜃 = tan−1 ( ) 1 + 𝑚2 𝑚1 Area of a Triangle by Coordinates Consider a polygon whose vertices have coordinates of (x1, y1), (x2, y2) and (x3, y3). The arrow shown in the figure moving in a counterclockwise direction indicates that the vertices must be written in the equation below in counterclockwise direction. 1 𝐴 = [(𝑥1 𝑦2 + 𝑥2 𝑦3 + 𝑥3 𝑦1 ) − (𝑦1 𝑥2 + 𝑦2 𝑥3 + 𝑦3 𝑥1 )] 2 Prepared by: Engr. Cabral, Engr. Celestial & Engr. Cruz Line – is defined as the shortest distance between two points. The following are the equations of the lines: A. General Equation: 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0 B. Point-Slope Form: (𝑦 − 𝑦1 ) = 𝑚 (𝑥 − 𝑥1 ) C. Slope-Intercept Form: 𝑦 = 𝑚𝑥 + 𝑏 D. Two-Point Form: 𝑦2 − 𝑦1 (𝑦 − 𝑦1 ) = ( ) (𝑥 − 𝑥1 ) 𝑥2 − 𝑥1 E. Intercept Form: 𝑥 𝑦 + =1 𝑎 𝑏 Prepared by: Engr. Cabral, Engr. Celestial & Engr. Cruz