VCE Specialist Mathematics Pocket Study Guide Contents Introduction 1 Vectors in two and three dimensions 2 Complex numbers 3 Coordinate geometry and sketch graphs 4 Circular functions 5 Antidifferentiation 6 Integration 7 Differential equations 8 Kinematics 9 Vector calculus 10 Dynamics iv 1 12 26 40 53 62 70 79 86 92 Introduction What do you really need to know for Specialist Mathematics? We’ve answered that question in this Pocket Study Guide. This handy guide gives you a summary of the entire course in an easy-to-use form. Keep it with you and look through it whenever you get the chance. The guide includes: • diagrams • formulae • graphics calculator and CAS advice • frequently asked questions • study tips • space for you to …show more content…
˜ ˜ ˜ ˜ ˜ ˜ • If a and b are perpendicular vectors, then ˜ ˜ a.b = | a| |b| cos 90° = 0. ˜ ˜ ˜ ˜ • If the scalar product of two non-zero vectors is zero, then the vectors are perpendicular. • If a = x1 i + y1 j + z1 k, and b = x2 i + y2 j + z2 k, then ˜ ˜ ˜ ˜ ˜ ˜ ˜ a.b = x1x2 + y˜ y2 + z1z2. 1 ˜ ˜ • The angle between two vectors is determined by a .b cos θ = ------------ . ˜ ˜ a b ˜ ˜ 6 1: Vectors in two and three dimensions 1.7 Resolution of vectors • The scalar resolute (or scalar projection) of a in the ˜ direction of b is the scalar product of a and unit ˜ ˜ a .b ˆ ˆ vector b, and is given by a.b = -------- . ˜b˜ ˜ ˜ ˜ ˜ a ~ b ~ • The vector resolute (or vector component) of a in ˜ ˆ )b = a .b b . ˆ -------the direction of b is given by ( a.b ˜b ˜2 ˜ ˜ ˜ ˜ ˜ ˜ a ~ ˆ ˆ a − (a.b)b ~~ ~ ˆ ˆ (a.b)b ~~ ~ b ~ • The vector resolute (or component) of a ˜ ˆ ˆ perpendicular to b is equal to a − ( a.b)b. ˜ ˜ ˜ ˜ ˜ 7 1: Vectors in two and three dimensions 1.8 Dependent and independent vectors • Two vectors a and b in the Cartesian plane are ˜ ˜ linearly dependent if and only if there are two y scalars x and y such that x a + yb = 0 , i.e. a = − -- b. x˜ ˜ ˜ ˜ ˜ Therefore, two vectors a and b are linearly ˜ ˜ dependent if and only if they are parallel. • Two vectors are said to be linearly independent if and only if the only solution to x a + yb = 0 is x = y = 0. ˜ ˜ ˜ • If a, b and c are three non-parallel vectors in the ˜ ˜
The catapult is a machine that launches an object into the air by using sudden tension. When people launched catapults, the object that was launched forms a parabola to hit the target otherwise known as a projectile. There are forces that act upon the object such as gravity and initial velocity. To calculate where the object lands, people use the vertex form and the quadratic formula. The vertex form is y = a(x – h)2 + k in which a is equal to the slope of the parabola, h is equal to half of the x value and k is equal to the y value of the maximum point. Also, people can find the farthest distance where an object could land by using the quadratic formula. Therefore, vertex form and quadratic formula are used to determine the path of the catapult’s
5.<HEF and <HED are adjacent complementary because they both are sharing point H and E and when those two angles are connecting the make 90
least five aspects, each one associated with one of the cardinal directions and the center. They
Select the best wording of the underlined part of the sentence. If you think the original sentence is best, choose answer A. The polarization
which the velocity is changing. Thus, when you look at the position of the dot at times shortly
In Kurt Lewin’s formula: B = f(P,E), P stands for ___ and E stands for ___.
Section B—DIRECTIONS: Complete each of the following sentences by writing in the Answers column the letter of the word or words that correctly completes each statement. (3 points for each correct answer)
8. Some of the key study areas are as follows: (while these are key areas remember that the exam is comprehensive for all the assigned course content and this study guide may not be all inclusive.
When being used for construction, the plumb bob is used to find the perpendicular: a straight line at a 90 degree angle to a given line or plane. To get the plumb line, end result, the plumb bob must hang freely, not touching any surface behind or below it. When the weight stops swinging, gravity pulls the weight to the center of the earth, leaving it directly under the point above it, resulting in a straight line. The perpendicular,
The overall upward or downward pattern of the data in an annual time series will be contained in the ____________ component.
When it comes to Euclidean Geometry, Spherical Geometry and Hyperbolic Geometry there are many similarities and differences among them. For example, what may be true for Euclidean Geometry may not be true for Spherical or Hyperbolic Geometry. Many instances exist where something is true for one or two geometries but not the other geometry. However, sometimes a property is true for all three geometries. These points bring us to the purpose of this paper. This paper is an opportunity for me to demonstrate my growing understanding about Euclidean Geometry, Spherical Geometry, and Hyperbolic Geometry.
Use the distributive property to remove the parentheses in the following expression. Then simplify your result if possible.
Euclid’s assumptions about his postulates have set the groundwork for geometry today. He provided society with definitions of a circle, a point, and line, etc and for 2000 was considered “the father of geometry.” His postulates proved to be a framework from which mathematics was able to grow and evolve, from two thousand years ago, till Newton and even to all our classrooms today.
If the load is applied at the mid- length a=b=L/2 then mid span deflection is: