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ENGR 1320 Final Review - Math • Major Topics: – Trigonometry – Vectors • Dot product • Cross product – Matrices • Matrix operations • Matrix equations • Gaussian Elimination – Complex numbers • Polar coordinates • Exponential form – Polynomials • Curve fitting • Roots – Calculus Trigonometry • We use 3 functions over and over again: sin(θ), cos(θ), tan(θ) • sin(θ) = y/r r • cos(θ) = x/r y • tan(θ) = y/x θ x • Example: Find the y component of this triangle: 10 pi/3 x y Vectors • Vectors represent a quantity in physical space with magnitude and direction – Knowing the magnitude and angle, trigonometry gives us the x and y components – It works the other way too: given magnitude the x and y components, we can θ find the magnitude and angle • Magnitude from pythagorian theorem • Angle from arc (or inverse) tangent x component • Example: What is the magnitude of this vector v v θ 10 5 y component Vector Notation • When representing vectors, we can either specify their magnitude and direction, or write them in components. The component method is generally more useful. We use unit vectors i and j to signify the x and y directions, respectively. So a vector that is three units in the x direction and 4 in the y direction would be written: v = 3i + 4j • Question: What is the magnitude of this vector? The angle with the x-axis? Vector addition • If we have 2 vectors v1 and v2, we can add them together by adding their components: 𝒗𝟏 = 3𝒊 + 4𝒋 𝒗𝟐 = 4𝒊 + 3𝒋 𝒗𝟏 + 𝒗𝟐 = 7𝒊 + 7𝒋 This is the ‘tip to tail’ method Vector Operations • There are 2 ways of multiplying vectors – Dot product • 𝒗𝟏 ∗ 𝐯𝟐 = v1 v2 cos 𝜃 • 𝒗𝟏 ∗ 𝐯𝟐 = v1 𝑥 𝑣2 𝑥 + v1 𝑦 𝑣2 𝑦 +v1 𝑧 𝑣2 𝑧 – Cross product 𝒊 • 𝒗𝟏 𝑥 𝐯𝟐 = 𝑣1 𝑥 𝑣2 𝑥 𝒋 𝑣1 𝑦 𝑣2 𝑦 𝒌 𝑣1 𝑧 𝑣2 𝑧 • See Vectors in MathCAD.pptx • Example: 𝒗𝟏 = 3𝒊 + 4𝒋 + 5𝒌 𝒗𝟐 = 1𝒊 + 2𝒋 + 3𝒌 What are the dot and cross products of these two vectors? Matrices • A matrix is a collection of values in structure. • Special matrix operations: – Transpose See matrix math – Determinant See Determinants and Adjoints – Inverse See matrix inverse Matrix Equations • We looked at several ways to solve the equation Ax = b. – Matrix inverse (similar to dividing by a matrix) – Gaussian elimination (similar to solving simultaneous algebraic equations) See Gaussian Elimination and More Gaussian Elimination 1 2 1 Example: solve 𝑥= using the matrix 2 −1 −1 inverse and gaussian elimination methods Complex numbers • We often find real applications that contain complex numbers in engineering – Vibrations in mechanics – Stability in control systems • We define the complex number i to be −1. – Question: what is i2? I3? • This result often arises in solving the quadratic formula: – What are the roots of 𝑥 2 + 1? See Introduction to Complex numbers Complex numbers • We can write complex numbers in 3 ways: – Components • Specify the real and imaginary components – Polar • Use trigonometry to convert into angles and magnitudes on the complex plane (Argand diagram) • See More complex numbers – Exponential • Taking the angle and magnitude from polar form, write the complex number as an exponential • See Exponential Form • Example: find the roots of 𝑥 2 − 𝑥 + 1 and write them in the 3 different forms Polynomials • A polynomial is an expression that follows the form: 𝑎0 𝑥 𝑛 + 𝑎1 𝑥 𝑛−1 + 𝑎2 𝑥 𝑛−2 + ⋯ + 𝑎𝑛−1 𝑥 + 𝑎𝑛 – This polynomial is nth order • How many roots does this polynomial have? • We used mathCAD and Matlab to find roots of polynomials of higher order than 2. • See Polynomials, Polynomials in MathCAD Calculus • We looked at a few basic concepts from calculus – Derivative • The slope of a curve at any point – Integral • The area under the curve at any point • We won’t be using MathCAD on the exam, so you will not be asked to solve equations with derivatives or integrals, but you might be asked questions on these general concepts. Study Strategy • Exam problems will be similar to homeworks • Several problems have been revisited in this class: – Electric circuits, truss equations, etc… • Look over the first 2 exams for representative problems (particularly the 1st for math-related problems)