Transcript ENGR 1320 Final Review

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)