- 3. Leading person of mathematicians. 1858 – 1932 Giuseppe Peano 1831 – 1916 Richard Dedekind 1953 Andrew Wiles 1499 – 1545 Chistoph Rudolff 288 a.C – 212 a.C Arquímides de Siracusa 1510 – 1558 Robert Recorde 200 – 284 Diofanto de Alejandría 1596 – 1650 Rene Descartes
- 4. Types of Numbers Numbers Naturals use to count and order the elements, 0 is the first element and the suspensive points mean that the natural numbers have infinite elements. Identify with this sign N The additions are ever closed Principle inductions Integers is an extension of natural Numbers. Identify with this sing z z- and z+ It is closed in add, subtractions and multiplications Rational Numbers Q could be represented in form of fractions or decimals. Exact. Only has one number after the decimal point Pure periodic. The amount after the decimal points is repeated in one or more numbers. Periodic mixed. One part is not exact, and one part is periodic or is repeated. Densidad de números racionales: existencia entre dos números racionales, de infinitos números racionales. Irrational Numbers the set of irrational numbers could be represent with letter I Nonrational Numbers Irrational Number cannot be expressed as a quotient of integers. Can be express in decimal form but the irrational Numbers neither terminate nor repeat. Real Numbers es la unión del conjunto de números racionales Q con el conjunto de números irracionales I, se representa con la letra R
- 5. Exponents: Potentiation of reals Numbers Is the operation that allow express in simplified form the multiplications of various equals factors. Properties or rules Product rule 𝑏𝑚 × 𝑏𝑛 = 𝑏𝑛+𝑚 add exponent and keep the base Quotient rule 𝑏𝑚 𝑏𝑛 = 𝑏𝑚−𝑛 subtract the exponent and keep the base Power rule 𝑏𝑚 𝑛 = 𝑏𝑚⋅𝑛 multiply the exponent keep the base Power of a product rule 𝑎𝑥𝑏 𝑛 = 𝑎𝑛 × 𝑏𝑛 Power of a quotient rule 𝑎 𝑏 𝑛 = 𝑎𝑛 𝑏𝑛 Exponentiation with negative exponent 𝑎 𝑏 −𝑛 = 𝑏 𝑎 𝑛 The elements Base and exponent = the answer is potentiation. Zero exponent all Number at 0 exponent is equal 1.
- 6. Radicals of reals Numbers Nth roots of a real Numbers is another real Numbers, if and if the nth roots is the same power. 𝑛 𝑎 = 𝑏 ↔ 𝑏𝑛 = 𝑎 Situation Index even and sub radical is even real number Index even and sub radical is negative not have answer Index odd and the sub radical is real Number positive Index odd and the sub radical is real Number negative Properties or rules Root of a product 𝑛 𝑎 × 𝑏 = 𝑛 𝑎 × 𝑛 𝑏 Root of a quotient 𝑛 𝑎 𝑏 = 𝑛 𝑎 𝑛 𝑏 Root of a root 𝑛 𝑚 𝑎 = 𝑛𝑥𝑚 𝑎 Root of a power 𝑛 𝑎𝑚 = 𝑎 𝑚 𝑛 Nth root of a positive Number to nth power 𝑛 𝑎𝑛 = 𝑎
- 7. Radicals of reals Numbers Simplifying expression with radicals Discompose in common factors Is simplified when the exponent in the sub radical are minor that the index. The GCD of the exponents of the factors of the sub radical quantity and of the index the root must be equal to 1 Applied the root properties of a product Some expression must simplify if the exponent in sub radical is higher or equal than the index.
- 8. Radicals of reals Numbers Arithmetic operation To add and subtract radicals simplify each radical and reduce like radicals. Multiplies radical with like index Multiplies the coefficient among if, and the amount of sub radicals, after simplify the result. Multiplies radicals with different index Reduce the radicals to radicals with same index. To calculate the common index use the following step Calculate the LCM among index of the radicals Divide the common index of each root and rise the sub radical amount of each result. Division of radicals Divide the coefficient among if and the sub radicals amount write inside of the common radical, simplify in the minor grade If the radicals are different convert to common radicals Rationalization Rationalization with monomials denominators Amplify the fraction in such a way that the radicals Will have exactly root. Rationalization with binomials denominators Amplify the fraction by the conjugate of the denominator and then simplify the expression.
- 9. Real Intervals Real Intervals is a subsets of the reals Numbers that contain all the Numbers understood among two points in a numerical line. The elements in intervals are infinite Classification Open : set of Numbers understood among point a and b not include a and b Close : set of Numbers understood among points a and b include a and b Semi open: set of Numbers formed by one of extreme and all the Numbers among a and b. Semi closed: set of Numbers formed by one of the extreme and all the Numbers among a and b Infinite: set of Numbers formed by all the reals Numbers higher or equals, lower or equal. Higher or lower that one of extreme.
- 10. Scientific Notation Both very large and very small Numbers frequently occurs in many fields of science. A positive Number is written in scientific notation if it is written as the product of a Number a, where 1 ≤ 𝑎 < 10, 𝑎𝑛𝑑 an integer power r of 10: 𝑎 × 10𝑟 To write a Number in scientific notation Move the decimal point in the original Number to the left or right so that the new Number has a value between 1 and 10 Count the Number of the decimal places the decimal point. If the original Number is greater than 10 is positive but is less tan 1 is negative. Multiply the new Number in step 1 by 10 raised to an exponent equal to the count found in step two Types of amounts Whole amount: put the point at the right of the amount of greater value and multiplies by 10 Decimal amount: displace the decimal point until it is located at the right of the positional greater amount Decimal amount with whole part zero: displace the point until it is located at the left of the first decimal place different of zero
- 11. ◦ Par ordenado ◦ Punto de par ordenada ◦ Vector del par ordenado ◦ En lugar de x se le llama componente real de z ◦ En lugar de y parte imaginaria del complejo z ◦ Suma de complejos 𝑎, 𝑏 + 𝑐, 𝑑 = 𝑎 + 𝑐, 𝑏 + 𝑑 ◦ Sumo las partes reales y aparte las partes imaginarias ◦ Producto de complejos 𝑎, 𝑏 ⋅ 𝑐, 𝑑 = 𝑎𝑐 − 𝑏𝑑, 𝑎𝑑 + 𝑏𝑐 ◦ Los complejos de la forma 𝑎, 0 = 𝑎 ◦ El complejo 0,1 = ⅈ defininción ◦ 𝑦 𝑒𝑠 𝑒𝑙 𝑟𝑒𝑎𝑙 𝑎, ⅈ 𝑒𝑠 𝑙𝑎 𝑑𝑒𝑓ⅈ𝑛ⅈ𝑐ⅈó𝑛 = 𝑦, 0 0,1 = 0 − 0, 𝑦 + 0 = 0, 𝑦 ◦ 𝑎, 𝑏 = 𝑎,0 0, 𝑏 = 𝑎 + 𝑏ⅈ ◦ ⅈ2 = ⅈ ⋅ ⅈ = 0,1 0,1 = 0 − 1,0 + 0 = −1,0 = ⅈ2 = −1 -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 Y-Values
- 12. Complex Numbers The equation 𝑥2 + 𝑎 = 0 where a is a real positive number, no have solution in the set of real numbers. But the square of a real number is no negative and when they are adding with other positive number their result it is no zero Imaginary Numbers are identified with letter i and is defined as ⅈ = −1 where ⅈ2 = −1 A complex Number is a Number can be written in the form 𝑎 + 𝑏ⅈ Where a and b are real Numbers. A complex Number that can be written in the form 0 + 𝑏ⅈ 𝑏 ≠ 0 is also called a pure imaginary number. Potentiation of complex Numbers The potentiation of unit imaginary is obtained to applied the properties of the potentiation and the definition of ⅈ 𝑒 ⅈ2 . ⅈ1 = ⅈ ⅈ2 = −1 ⅈ3 = ⅈ × ⅈ2 = ⅈ × −1 = −ⅈ ⅈ4 = ⅈ2 × ⅈ2 = −1 𝑥 −1 = 1 The exponent is expressed like 𝑎4𝑛+𝑟 where n is a integer number and r is a integer positive number minor that 4 Then applied the properties if potentiation. Set of complex Numbers 𝐶 = 𝑎 + 𝑏ⅈ: 𝑎, 𝑏 ∈ 𝑅, ⅈ = −1 A is real part and bi is imaginary part 𝑅 ⊂ 𝐶
- 13. Arithmetical operation Addition and subtraction add and subtract the real parts and the imaginary parts 𝑧 ± 𝑤 = 𝑎 ± 𝑐 + 𝑏 ± 𝑑 ⅈ Multiplication Apply the distributive property Solve the power of i Reduce like terms 𝑧𝑥𝑤 = 𝑎𝑐 − 𝑏𝑑 + 𝑎𝑑 + 𝑏𝑐 ⅈ Division Multiplies the dividend and the divisor by the conjugant of the divisor. Solve the indicated operation. Complex Numbers
- 14. First grade equations in one variable Equation Is mathematical statement that two expression have equal value. Members: expression that have at each side of the equal. The expression in the right side is second term and the expression in the left side is first term Incognita or variable unknown value Terms: are separate by sing like add or subtract, is impossible use time and divide Grade relative add of exponent Grade absolute is the greater exponent of the variable Independent term: no have variable Solution Is the value that could take the variable for the equality is real.
- 15. Isolate of variable Transposition of term Change term of a member to other without the equation change. Other form is duplicate the term in both members with different sing Isolate the variable is that the variable is alone. Ever do you could look that the term is divide at change is multiplies and vice versa.
- 16. First grade inequations in one variable Is an inequality that can be written in the form 𝑎𝑥 + 𝑏 < 𝑐 where a, b and c are real numbers and a in not 0 Indicate that one amount is greater o les the other amount properties If add or subtract the same Number in both members the inequality is conservator. If multiplies or divide for a positive Number both members. The inequality is conservator. if multiplies or divide for a negative Number both members. Change the inequality. Solution Calculate the values of the variable to allow the inequality Realize the same steps of the equalities Apply the properties of the inequalities Represent with a real interval.
- 17. Second grade equations 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 c𝑜𝑛 𝑎, 𝑏, 𝑐 ∈ 𝑅 𝑦 𝑎 ≠ 0 incomplete Are all where b=0 ó c=0 All quadratic equations have two reals root different, two complex root different or only one real root. 𝑎𝑥2 = 0 have one solutions 𝑎𝑥2 ± 𝑏𝑥 = 0 have two solutions 𝑎𝑥2 ± 𝑐 = 0 have two solutions Complete Are all where 𝑏 ≠ 𝑂 𝑦 𝑐 ≠ 0 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 Solution by factorization Write the equation in form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 Factorization if is possible Equality to 0 Solve each equations of first grade to calculate the solutions General form 𝑥 = −𝑏 ± 𝑏2 − 4𝑎𝑐 2𝑎 Write the equations in this form 𝑎𝑥2 + 𝑏𝑥 + 𝑐 = 0 Substitute the values of a, b, c Realize the operations indicated in the general form.
- 18. Cartesian plane Two numerical lines called axes Axis x is abscissa Axis y is ordinates Have four quadrants Use a coordinates 8 3.2 -8 -10 -8 -6 -4 -2 0 2 4 6 8 10 12 -10 -5 0 5 10 1st quadrant 2nd quadrant 3rd quadrant 4th quadrant
- 19. -6 -4 -2 0 2 4 6 -5 -4 -3 -2 -1 0 1 2 3 4 5 Axis Title Axis Title Y-Values
- 20. Distance between two points The distance between two points is the length of the line segment that joins sayings points When the points are locates only in abscissas axis or in ordinates axis The absolute value is the difference between the abscissas and ordinates Use a Pythagoras theorem 𝑃𝑄 2 = 𝑃𝑅 2 + 𝑄𝑅 2 𝑑2 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2 𝑑 = 𝑥2 − 𝑥1 2 + 𝑦2 − 𝑦1 2
- 21. Unit 2 line equations and equations system
- 22. Line Equations The line equation of first grade have the form 𝑦 = 𝑚𝑥 + 𝑏 Where m represent the slope b the ordinate or intercepted whit the axis y The slop in the line is defined as the ratio between the vertical variation and horizontal variation. General equation 𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0
- 23. Parallel line equation Two lines could be Parallels perpendiculars Parallels secants 𝑙1 = 𝑦1 = 𝑚1𝑥1 + 𝑏1 𝑙2 = 𝑦2 = 𝑚2𝑥2 + 𝑏2 To find the explicit equation of a parallel line If is expressed in explicit form only identify the slope In other form use a formula 𝑚 = − 𝐴 𝐵 Apply the procedure to determine the line equation. You could write a parallel line equation through the explicit equation or slope point equation.
- 24. Perpendicular line equations If the lines are cut in one point are called a secant This secant could be called perpendicular If possible, when form a right angle And if the product of their slopes are equal to -1 𝑚1 = − 𝐴 𝐵 𝑚2 = − 𝐴 𝐵 𝑚1𝑥𝑚2
- 25. 2 x 2 System only have two equations and two variables or unknowns. Methods to solve 2 x 2 System Graphic: is graphic representation of the lines that correspond to the equation system Lines that cut in alone point, include that the system have only one answer for this reason are called Determinate or consistent. Lines that match in all points, have infinite solutions and are called indeterminate Parallel not have solutions, and not have match points, are called inconsistent. Substitution need three steps Isolate one of the variable in anyone of the equations system Substitute the resultant expression in the other equations and isolate the remaining variable The obtained value in the 2-step substitute in anyone of the equations or in the first isolate equations Equalization Isolate the variable in both equations Equalizes the obtained expression and isolate the variable This value is substitute in anyone of the equations. Elimination or reductions Simplify both equations in only one Amplify one of the equations in such a way that the coefficient that variables are opposite. Finally substitute the value of the variable in anyone of the equations. Inconsistent Indeterminant Consistent Determinants or Cramer's Associated Number to an array to rows and columns The determinant value is equivalent to at the difference between the product of the principal diagonal and the product of the secondary diagonal.
- 26. Equations system with three variable first grade 3x3 System have three equations and three variables or unknowns Substitution Isolate one of the variables in anyone of the equations Substitute the isolate expression in the remaining equations realize the operations Forming a 2 x 2 system Select an equations and substitute the values of the obtained variables Method of elimination, add and subtractions Select two equations anyone Select one variable to eliminate. Amplify the coefficient value in one equations with opposite sing Repeat the step select one equations and the remaining equations. You have a 2 x 2 System solve the system Select an equations and substitute the values of the obtained variables Method of Cramer o method of determinant Work with the coefficient Method of Sarrus o cofactor method Work with the coefficient
- 27. Problem solving Solutions order Understand the data of the problem Design the plan Develop the plan Check t the answer Write the answer
- 29. Angles the space (measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet. In other words, is amplitude of rotation of a semi-straight on their origin. Sexagesimal system have a Sexagesimal degree is 1 360 parts of angle generate for a complete rotation of one side. Minutes Seconds Conversion degree to sexagesimal degree Multiplies the decimal part by 60 Multiplies the decimal part of the result in first step Divide minutes by 60 Divide second by 3600 Measure of angles with protractor Protractor is divided by sexagesimal degrees Central angle with center in the origin and radio is that formed to radios the unit of measure is radian Classifications of angles Angle in normal position initial side match with positive semiaxis and their vertex is in the origin. Quadrantal angle is the angle whose initial side is in the positive semiaxis, and the final side is localized in one of the semiaxis of the cartesian plane Coterminal angle are when their initial sides and final sides are the same. Regardless magnitude and direction. In cartesian plane the final side could realize more tan one turn in negative or positive direction Reference angle acute angle that form the final side of the angle with one of the semiaxis of axis x If the final side are in the second quadrant their reference angles need this formula 𝜃𝑅 = 180° − 𝜃 If the final side are in the third quadrant their reference angle need this formula 𝜃𝑅 = 𝜃 − 1800 If the final side are in the fourth quadrant their reference angle need this formula 𝜃𝑅 = 3600 − 𝜃
- 30. ℎ2 = 𝑎2 + 𝑏2 ℎ = 𝑎2 + 𝑏2 Opposite cateto is situated in front of the given angle formula 𝑎2 = 𝑐2 − 𝑏2 = 𝑎 = 𝑐2 − 𝑏2 Adjacent cateto is situated next to the given angle and that is not the Hypothenuse formula 𝑏2 = 𝑐2 − 𝑎2 = 𝑏 = 𝑐2 − 𝑎2 The square of the length of the hypothenuse is equal to add of the squares of the lengths of the hick.
- 31. To solve rectangle triangle Unknowns the measures of one of the sides of acute angle Use a trigonometric ratio depend on the side that you known Known the length of the sides Use inverse trigonometric ratio to calculate the measure of the inner angle unknowns Use a Pythagoras Theorem Or use a trigonometric ratio that relation the length with the unknown side with the measure of one of the sides and the given angle
- 32. Trigonometric functions in the plane Trigonometric functions in the plane Unit Circle Center is in the origin point of the cartesian plane and whose radius is equal to 1 The equations related is 𝑥2 + 𝑦2 = 1 The distance of all point that belongs at unit circle of the cartesian plane is equal to 1 Relation between trigonometric functions and the unit circle Build and angle Ɵ ⅈn normal position, whose final side intersects the unit circle at a point 𝑅 𝑥, 𝑦 Trace a perpendicular segment at x axis to form rectangle triangle. Se establecen funciones trigonométricas de la siguiente forma 𝑠𝑒𝑛𝜃 = 𝑦 1 = 𝑦 tan 𝜃 = 𝑦 𝑥′ 𝑥 ≠ 0 s𝑒𝑛𝜃 = 1 𝑥′ 𝑥 ≠ 0 cos 𝜃 = 𝑥 1 = 𝑥 csc 𝜃 = 1 𝑦′ 𝑦 ≠ 0 cot 𝜃 = 𝑥 𝑦′ 𝑦 ≠ 0 Domain of the trigonometric functions The domain of the trigonometric functions depends on the coordinates 𝑥, 𝑦 𝑎𝑛 𝑑 th 𝑝𝑜ⅈ𝑛𝑡 𝑅 𝑥, 𝑦
- 33. Trigonometric functions of quadrantal angles Trigonometric functions of quadrantal angles Values of the trigonometric functions for quadrantal angles of 0° and 360° The final side of the quadrantal angles of 0 and 360 coincide, and the value of trigonometric functions are the same. cos = 𝜃 = 𝑥 = 1 𝑠𝑒𝑛𝜃 = 𝑦 = 0 tan 𝜃 = 𝑦 𝑥 = 0 sec 𝜃 = 1 𝑥 = 1 1 = 1 cot 𝜃 = 𝑥 𝑦 = 1 0 𝑛𝑜 𝑑𝑒𝑓ⅈ𝑛ⅈ𝑑𝑜 csc 𝜃 = 1 𝑦 = 1 0 𝑛𝑜 𝑑𝑒𝑓ⅈ𝑛ⅈ𝑑𝑜 Values of the trigonometric functions for quadrantal angle of 90° The point of intersection of the unit circle with the final angle of 90° is (0,1) cos 900 = 𝑥 = 0 𝑆e𝑛900 = 𝑦 = 1 tan 900 = 𝑦 𝑥 = 1 0 𝑛𝑜 𝑑𝑒𝑓ⅈ𝑛ⅈ𝑑𝑜 sec 900 = 1 𝑥 = 1 0 𝑛𝑜 𝑑𝑒𝑓ⅈ𝑛ⅈ𝑑𝑜 csc 900 = 1 𝑦 = 1 1 = 1 cot 900 = 𝑥 𝑦 = 0 1 = 0 Values of the trigonometric functions for quadrantal angle of 180° The point of intersection of the unit circle with the final side of the angle of 180° is (-1,0) Cos 1800 = 𝑥 = 1 tan 180° = 𝑦 𝑥 = 0 −1 = 0 Sen 180°= y = 0 sec 180° = 1 𝑥 = 1 −1 = −1 Csc 180°= 1 𝑦 = 1 0 no defined cot 180°= 𝑥 𝑦 = −1 0 no defined Values of the trigonometric functions for quadrantal angle of 270° Cos 270° = x = 0 sec 270° = 1 𝑥 = 1 0 no definido Sen 270° = y = -1 Csc 270° = 1 𝑦 = 1 −1 = −1 Tan 270° = 𝑦 𝑥 = −1 0 𝑛𝑜 𝑑𝑒𝑓ⅈ𝑛ⅈ𝑑𝑜 Cot 270° = 𝑥 𝑦 = 0 −1 = 0
- 34. Trigonometric functions of special angles Special angles 30 45 60
- 36. Sine law Sine rule Study the triangle ABC shown below. Let B stands for the angle at B. Let C stand for the angle at C and so on. Also, let b = AC, a = BC and c = AB
- 37. COSINE LAW
- 39. Trigonometric identities Una identidad trigonométrica es una igualdad entre razones trigonométricas que son verdades para todo valor de la variable (ángulo) que intervenga en dicha igualdad. Dichas identidades se
- 40. (74) Identidades Trigonométricas Fundamentales - Clase - YouTube
- 44. Vectors in the plane ◦ un vector no es más que un trozo de recta, en el que se diferencia claramente su origen y su extremo. ◦ Un vector es un segmento orientado con un punto inicial u origen y uno final a extremo que termina con una flecha. ◦ DEFINICIÓN: COMPONENTES DE UN VECTOR. Son dos valores que vienen dados en forma de par de números, los cuáles indican las unidades que tenemos que desplazarnos horizontalmente y verticalmente respectivamente, para llegar desde el origen del vector al extremo de éste. ◦ 𝐴𝐵 ◦ CARACTERÍSTICAS DE UN VECTOR. MÓDULO DIRECCIÓN Y SENTIDO ◦ Magnitud o Módulo: Es el tamaño que tiene el segmento orientado. ◦ Dirección: Es la inclinación que tiene el vector respecto al eje de abscisas ( eje de las X). ◦ Esta inclinación se mide a través del ángulo menor que forma el vector con el eje OX ó un eje paralelo a éste. ◦ Sentido: Es la orientación que adopta el vector. Podemos diferenciar entre Norte, Sur, Este, Oeste, Noreste, Noroeste, Sureste, Suroeste.
- 48. ◦ 𝑣𝑥 = 𝑣 cos 𝜃 ◦ 𝑣𝑦 = 𝑣 𝑠𝑒𝑛𝜃 ◦ 𝑣𝑥 = 23 cos 250 = ◦ 𝑣𝑥 = 23(0.90630778703664996324255265675432)= ◦ 𝑣𝑥 = 20.845079101842949154578711105349 ◦ 𝑣𝑦 = 23 sⅈn 25 ◦ 𝑣𝑦 = 23 (0.42261826174069943618697848964773) ◦ 𝑣𝑦 = 9.7202200200360870323005052618978 Formulas to obtain the components
- 49. COORDENADAS POLARES Y RECTANGULARES
- 51. ¿Qué son las coordenadas cartesianas? Antes de saber para qué sirven las coordenadas cartesianas, vamos a explicar qué son. Las coordenadas cartesianas, también conocidas como coordenadas rectangulares, son un tipo de coordenadas ortogonales que se usan en espacios euclídeos. Las coordenadas cartesianas se utilizan para representar gráficamente una relación matemática o movimiento en posición física. Además, se caracterizan por tener como referencia unos ejes ortogonales que concurren entre sí en el llamado punto de origen.
- 52. ◦ 𝑟 𝑥2 + 𝑦2 𝜃 = tan−1 𝑦 𝑥 ◦ ℎ𝑥 = ℎ cos 𝜃 ℎ𝑦 = ℎ sⅈn 𝜃 DE RECTANGULARES A POLARES Y VICEVERSA 7, −5 𝑟 = 7 − 5 2 𝑟 = 49 + 25 𝑟 = 74 𝜃 = tan−1 −5 7 𝜃 = tan−1 ( − 0.71428571428571428571428571428571) 𝜃 = −35.537677791974382608859929458258 𝜃 = 20.56 − 180 = 159.44 ℎ = 73 cos 159.44 ℎ = 8.5440037453175311678716483262397 −0.93630493941540936613332849307122 ℎ = −7.9997929091245617358478026812182 ℎ𝑦 = 73 𝑠𝑒𝑛159.44 ℎ𝑦 = 8.5440037453175311678716483262397 0.35118806988037990068702574268117 ℎ𝑦 = 3.0005521843688007314643830919544
- 53. Operations with vectors ◦ 𝑎 + 𝑏 = 𝑥1, 𝑦1 + 𝑥2, 𝑦𝑧 ◦ 𝑎 − 𝑏 = 𝑥1, 𝑦1 − 𝑥2, 𝑦2 ◦ Type equatⅈon here.
- 56. ◦ Para calcular la magnitud se utiliza la siguiente formula: ◦ |AB| = √ [ (ABx)2 + (ABy)2]
- 57. Producto punto (producto escalar, producto interior) Módulo de un vector El módulo de un vector es un número siempre positivo y solamente el vector nulo tiene módulo cero. Para calcular el modulo utilizamos la siguiente formula: Angulo entre dos vectores Para hallar el ángulo entre dos vectores utilizamos la siguiente formula:
- 58. Point product(producto escalar, producto interior) ◦ 𝐷𝑜𝑠 𝑑ⅈ𝑚𝑒𝑛𝑠ⅈ𝑜𝑛𝑒𝑠 𝑢 = 𝑎, 𝑏 𝑣 = 𝑐, 𝑑 ◦ 𝜈 ⋅ 𝑣 = 𝑎𝑐 + 𝑏𝑑 Producto punto El producto punto o producto escalar de dos vectores es una operación que da como resultado un número real. Hay distintas formas de definir esta operación, una de ellas es por medio de multiplicar el producto de los módulos de los vectores por el coseno del ángulo que forman, esto es 𝑇𝑟𝑒𝑠 𝑑ⅈ𝑚𝑒𝑛𝑠ⅈ𝑜𝑛𝑒𝑠 𝑈 = 𝑎1, 𝑎2, 𝑎3 𝑣 = 𝑏1, 𝑏2, 𝑏3 𝜈 ⋅ 𝑣 = 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3
- 59. Properties ◦ 𝑎. 𝑎 = 𝑎 2 el producto punto de un vector 𝑎 multiplicado por si mismo es igual a la norma de ese mismo vector elevado al cuadrado. ◦ 𝑎 𝑎1𝑏, 𝑐 𝑎 ⋅ 𝑎 = 𝑎, 𝑏, 𝑐 𝑎, 𝑏, 𝑐 = 𝑎2 , 𝑏2 , 𝑐2 = 𝑎 2 𝑎 = 𝑎2, 𝑏2, 𝑐2 𝑎 2 = 𝑎2, 𝑏2, 𝑐2 2 𝑎 2 = 𝑎2, 𝑏2, 𝑐2 ◦ 𝑎 ⋅ 𝑏 = 𝑏 ⋅ 𝑎 = 𝑎, 𝑎, 𝑎 𝑏, 𝑏, 𝑏 Propiedad conmutativa ◦ 𝑎 ⋅ 𝑏 = 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3 ◦ 𝑏 ⋅ 𝑎 = 𝑏1𝑎1 + 𝑏2𝑎2 + 𝑏3𝑎3
- 60. Properties ◦ 𝑎 𝑏 + 𝑐 = 𝑎 ⋅ 𝑏 + 𝑎 ⋅ 𝑐 El producto punto con la suma de vectores es igual a distribuir el producto punto por el vector más el producto punto por el vector dos. ◦ 𝑘𝑎1 ⋅ 𝑏 = 𝑘 𝑎 ⋅ 𝑏 = 𝑎 ⋅ 𝑘𝑏 𝑎, 𝑎, 𝑎 𝑏, 𝑏, 𝑏 ◦ 𝑘𝑎 ⋅ 𝑏 = 𝑘𝑎1 , 𝑘𝑎2 , 𝑘𝑎3 . 𝑏1, 𝑏2, 𝑏3 = 𝑘𝑎1𝑏1 + 𝑘𝑎2𝑏2 + 𝑘𝑎3𝑏3 ◦ 𝑘 𝑎1𝑏1 + 𝑎2𝑏2 + 𝑎3𝑏3 ◦ 𝑘 𝑎 ⋅ 𝑏 ◦ 0 ⋅ 𝑏 = 0 vector cero con el producto punto por cualquier vector es igual al escalar cero, en dos dimensiones el vector es (0,0) y en tres dimensiones es (0,0,0)
- 61. Unit vectors ◦ 𝑣 = 𝑣 = 1 ◦ Se denomina vector unitario a aquel vector que tiene por módulo la unidad. Podemos calcular un vector unitario en la dirección del vector sin más que dividir por su módulo 𝑣 = 3 5 , 4 5 = 𝑣 = 3 5 2 + 4 5 2 𝑣 = 9 25 + 16 25 25 25 = 1 Sí me encuentro con un vector que no es Unitario en la misma dirección que el vector Unitario, para obtener la respuesta solo dividimos el vector por la norma del vector, y ahí obtendremos un vector unitario. 𝑣 𝑣
- 62. Unit vectors ◦ 𝑣 = −2,6 𝑣 −2,6 2 ◦ 𝑣 = 4 + 36 𝑣 = 40 ◦ 𝑣 𝑣 = − 2 40 , 6 40 = −2 447 , 6 4.47 = −0.45,1.34 ◦ 𝑣𝑒𝑐𝑡𝑜𝑟𝑒𝑠 𝑢𝑛ⅈ𝑡𝑎𝑟ⅈ𝑜𝑠 𝑑𝑒 𝑏𝑎𝑠𝑒 𝑐𝑎𝑛ó𝑛ⅈ𝑐𝑎 𝑑𝑜𝑠 𝑑ⅈ𝑚𝑒𝑛𝑠ⅈ𝑜𝑛𝑒𝑠 𝐼 = 1,0 𝐽 = 0,1 ◦ 𝑣𝑒𝑐𝑡𝑜𝑟𝑒𝑠 𝑢𝑛ⅈ𝑡𝑎𝑟ⅈ𝑜𝑠 𝑑𝑒 𝑏𝑎𝑠𝑒 𝑐𝑎𝑛ó𝑛ⅈ𝑐𝑎 𝑡𝑟𝑒𝑠 𝑑ⅈ𝑚𝑒𝑛𝑠ⅈ𝑜𝑛𝑒𝑠 𝐼 = 1,0,0 𝐽 = 0,1,0 𝑘 = 0,0,1
- 63. ◦ 2.32,4.098 ⋅ 𝜈 = 2.32,4.098 ◦ 𝑣 = 5.3824 + 16.793604 ◦ 𝑣 = 22.176004
- 66. Angle between vectors ◦ Ángulo de dos vectores Dados dos vectores u , v , si elegimos un origen común para ambos, se llama ángulo de dos vectores al menor de los ángulos que forman. Este ángulo, como ya sabemos, cumple el siguiente convenio: positivo si avanzamos en sentido antihorario, y negativo en sentido horario.
- 68. ANGLE BETWEEN VECTORS 𝐴 = 4,3 𝐵 = 5, −2 𝐴 ⋅ 𝐵 = 4,3 ⋅ 5, −2 𝐴 ⋅ 𝐵 = 4 5 + 3 −2 𝐴 ⋅ 𝐵 = 20 − 6 𝐴 ⋅ 𝐵 = 14 𝐴 = 4 + 3 2 𝐴 = 16 + 9 𝐴 = 25 𝐴 = 5 𝐵 = 5 + −2 2 𝐵 = 25 + 4 𝐵 = 29 cos 𝜃 = 𝐴 ⋅ 𝐵 𝐴 𝐵 cos 𝜃 = 14 5 29 cos 𝜃 = 14 5 5.3851 cos 𝜃 = 14 12.0415 cos 𝜃 = 0.51995 𝜃 = cos−1 0.51995 𝜃 = 58.6711
- 71. PRODUCTO NOTABLE
- 74. Proyección ortogonal de un vector ◦ Pro𝑦𝑣𝑈 o cuál quiere decir que estamos ante la proyección de u sobre v ◦ Pro𝑦𝑣𝑈 = 𝑣⋅𝑈 𝑣 2 𝑣 ◦ Pro𝑦𝑣𝑈 = comp𝑣𝑈
- 76. Matrices 𝑎 𝑎 𝑎 𝑎 columna fila 𝐴 = 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝐶 = 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 𝑎 2 columnas 2 filas 3 columnas 3 filas 5columnas 3 filas Matriz de tamaño 2x2 Matriz de tamaño 3x3 Matriz de tamaño 3x5 Siempre serán nombradas con la letra mayúscula.
- 77. ◦ Las matrices pueden tener entradas en ℝ 𝑜 ℂ 𝑜 ℚ etc. ◦ 𝐴 = 2 + ⅈ −3ⅈ 4 7ⅈ 0 5 − 7ⅈ 𝐴 ∈ 𝑀2×3 ℂ Matrices
- 78. Matrices ◦ 𝑎11 𝑎12 … 𝑎ln 𝑎21 𝑎22 … ⋮ ⋮ ⋱ 𝑎𝑚1 𝑎𝑚2 … 𝑎2𝑛 ⋮ 𝑎𝑚𝑛 ◦𝑎11 este es el elemento de la fila uno columna uno. Vector fila Vector Columna Usan letras como la 𝑣𝑦𝑢
- 79. ◦ SUMA Y RESTA O ADICIÓN Y SUSTRACCIÓN ◦ Para sumar y restar ambas deben ser del mismo tamaño o tener la misma dimensión, de no suceder así no se podría resolver la ecuación. ◦ Producto ◦ Podemos multiplicar cualquier número por la matriz ◦ Sólo debemos multiplicar el escalar por cada número de toda la matriz que se nos presente ◦ Igualdad de matrices ◦ Las matrices serán like term si y sólo si tiene ◦ Misma dimensión y los mismos elementos. ◦ DIVISION DE MATRICES ◦ Determinar el inverso multiplicativo del numero real es posible determinar la división
- 80. PROPIEDADE S
- 81. Multiplicación ◦ A= 2 1 −4 −1 −3 4 4 3 −2 Por B= 2 0 1 −3 5 1 ◦ 2 2 + 1 1 + −4 5 = 4 + 1 − 20 = −15 ◦ 2 0 + 1 −3 + −4 1 = 0 − 3 − 1 = −7 ◦ −1 2 + −3 1 + 4 5 = −2 − 3 + 20 = 15 ◦ −1 0 + −3 −3 + 4 1 = 0 + 9 + 4 = 13 ◦ 4 2 + 3 1 + −2 5 = 8 + 3 − 10 = 1 ◦ 4 0 + 3 −3 + −2 1 = 0 − 9 − 2 = −11 ◦ −15 −7 15 13 1 −11
- 82. DETERMINANTE
- 90. Matriz de cofactores transpuesta
- 92. Matriz Inversa 𝐴 = 𝑎 𝑏 𝐶 𝑑 𝐴−1 = 𝑑 − 𝑏 −𝑐 𝑎 𝐴 𝐴 ⋅ 𝐴−1 = 1 𝐴 ⋅ 𝐴−1 = 𝑎𝑝 + 𝑏𝑟 𝑐𝑞 + 𝑏𝑠 𝑐𝑝 + 𝑑𝑟 𝑐𝑞 + 𝑑𝑠 = 1 0 0 1 𝑏𝑢𝑠𝑐𝑎𝑚𝑜𝑠 𝑒𝑙 𝑣𝑎𝑙𝑜𝑟 𝑑𝑒 𝑝, 𝑞, 𝑟, 𝑠 𝑎𝑝 + 𝑏𝑟 = 1 𝑃 = 1 − 𝑏𝑟 𝑎 𝑐𝑞 + 𝑏𝑠 = 0 𝑐𝑝 + 𝑑𝑟 = 0 𝐶 1 − 𝑏𝑟 𝑎 + 𝑑𝑟 = 0 𝑐𝑞 + 𝑑𝑠 = 1
- 93. 𝑃 = 1 − 𝑏𝑟 𝑎 𝑃 = 1 − 𝑏 −𝑐 𝑎𝑑 − 𝑏𝑐 𝑎 𝑃 = 1 + 𝑏𝑐 𝑎𝑑 − 𝑏𝑐 𝑎 𝑃 = 𝑎 𝑑 − 𝑏𝑐 + 𝑏𝑐 𝑎𝑑 − 𝑏𝑐 𝑎 𝑃 = 𝑎 𝑑 𝑎𝑑 − 𝑏𝑐 𝑎 = 𝑎 𝑑 𝑎 𝑎𝑑 − 𝑏𝑐 = 𝑑 𝑎𝑑 − 𝑏𝑐 𝐶 1 − 𝑏𝑟 𝑎 + 𝑑𝑟 = 0 𝑐 1 − 𝑏𝑟 𝑎 + 𝑑𝑟𝑎 𝑎 = 0 𝑎 𝑎 𝑐 − 𝑐𝑏𝑟 + 𝑑𝑟𝑎 = 0 𝑐 + 𝑑𝑎 − 𝑏𝑐 𝑟 = 0 𝑟 = −𝑐 𝑎𝑑 − 𝑏𝑐 = − 𝑐 𝑎𝑑 − 𝑏𝑐
- 94. 𝑐𝑞 + 𝑏𝑠 = 0 𝑐𝑞 + 𝑑𝑠 = 1 𝑞 = −𝑏𝑠 𝑎 𝑐 −𝑏𝑠 𝑎 + 𝑑𝑠 = 1 𝑐 −𝑏𝑠 𝑎 + 𝑑𝑠𝑎 𝑎 = 1 𝑎 ⋅ 𝑎 −𝑏𝑐𝑠 + 𝑑𝑠𝑎 = 𝑎 −𝑏𝑐 + 𝑑𝑎 𝑠 = 𝑎 𝑠 = 𝑎 𝑎𝑑 − 𝑏𝑐 𝑐𝑞 − 𝑑𝑠 = 1 𝑐𝑞 + 𝑑 𝑎 𝑎𝑎 − 𝑏𝑐 = 1 𝑐𝑞 + 𝑑𝑎 9𝑑 − 𝑏𝑐 = 1 𝑐𝑞 = 1 − 𝑑𝑎 𝑎𝑑 − 𝑏𝑐 𝑐𝑞 = 𝑎 𝑑 − 𝑏𝑐 − 𝑑𝑎 𝑎𝑑 − 𝑏𝑐 𝑐𝑞 = −𝑏𝑐 𝑎𝑑 − 𝑏𝑐 𝑞 = −𝑏𝑐 𝑐 𝑎𝑑 − 𝑏𝑐 𝑞 = −𝑏 𝑎𝑑 − 𝑏𝑐 𝑞 = − 𝑏 𝑎𝑑 − 𝑏𝑐
- 95. ◦ PROPIEDADES DE LA MATRIZ INVERSA ◦ 1. La matriz inversa si existe es única. ◦ 2. (A-1)-1 = A, es decir, la inversa de la inversa es la matriz inicial. ◦ 3. (A·B)-1 = B-1·A-1 ◦ 4. |A-1| = 1 / |A|