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Specialist Maths Study Guide

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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 ˜ ˜

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