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 cos(θ) = x/r tan(θ) = y/x Example: Find the y component of this triangle: x y r θ x y 10 pi/3

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 the x and y components, we can find the magnitude and angle Magnitude from pythagorian theorem Angle from arc (or inverse) tangent Example: What is the magnitude of this vector v x component y component magnitude θ 10 5 v θ

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

Vector Operations

Matrices A matrix is a collection of values in structure. Special matrix operations: – Transpose See matrix mathmatrix math – Determinant See Determinants and AdjointsDeterminants and Adjoints – Inverse See matrix inversematrix inverse

Matrix Equations

Complex numbers

Polynomials

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 1 st for math-related problems)