Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
A horizontal ciothesiine is tied between 2 poles, 21! meters apart.
When a mass of 3 kilograms is tied to the middle of the clothesline, it sags a distance of 1 meters.
What is the magnitude of the tension on the ends of the clothesline?
Ne TE: Use g 2 9.
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
6. Vectors
For purposes of applications in calculus and physics, a vector has both a direction and a magnitude
(length), and is usually represented as an arrow. The start of the arrow is the vectors foot, and the
end is its head. A vector is usually label
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
Vectors
Vectors
head
A vector is a quantity that has both magnitude
and direction.
Geometrically a vector can be represented by an
arrow. The magnitude is the length of the vector
and the arrow points in the direction of the
vector.
We denote a vector by
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
The Cross Product
The Cross Product
If = 1 , 2 , 3 and = 1 , , 3 then the cross product of a and b is
= 2 3 3 2 1 3 3 1 + 1 2 2 1
The cross product is a vector.
The formula seems impossible to remember, but it becomes easy if we use
determinants:
A det
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
13. Planes in
Given a point 0 = (0 , 0 , 0 ) and a vector = , , , a plane that passes through 0 and is
normal (orthogonal) to n has the equation
( 0 ) + ( 0 ) + ( 0 ) = 0,
which simplifies to + + = , where = 0 + 0 + 0 .
Example 13.1: Find the equation of
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
Proof of the Magnitude of a Cross Product,   =  sin
Let = , , and = , , be two vectors in , and let be the angle that the two vectors form when their
feet are placed together. The cross product of u and v is
=
=
+
= , , .
Therefore, the ma
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
The Travel In One Direction Then Make One Right Angle
Turn To Get To Desired Finish Point Problem
Consider the following problem:
You are at point A = (2, 3) and start walking in the direction of = 1,4. Your finish point will be B =
(5,3). You are allowed
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
1. The xyz Coordinate Axis System
The xyz coordinate axis system is denoted 3 , and is represented by three real number lines
meeting at a common point, called the origin. The three number lines are called the xaxis, the yaxis, and the zaxis. Together,
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
The Dot Product
The Dot Product
If a a1, a2 , a3 and b b1, b2 , b3 , then the dot product of a and b is
a b a1b1 a2b2 a3b3
The dot product is a scalar (a number) and it is also called the scalar product
or the inner product.
Example: Let = 5, 0, 2 and = 3
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
3D Coordinate Systems
3D Coordinate Systems
A threedimensional coordinate system consists
of three lines perpendicular to each other and
through a point O called the origin.
These three lines are called the coordinate axes
and labeled the xaxis, yaxi
Arizona State University at the Polytechnic Campus
Calculus for Engineers III
MATHEMATIC 267

Spring 2016
12. Lines in
Given a point 0 = (0 , 0 , 0 ) and a direction vector 1 = , , in 3 , a line that passes
through 0 and is parallel to is written parametrically as a function of :
() = 0 + ,
() = 0 + ,
() = 0 +
Using vector notation, the same line is written
Arizona State University at the Polytechnic Campus
Elementary Statistics
MATHEMATIC 226

Summer 2016
8/7/2012
Graphical Analysis
Graphical Analysis
Useful to identify
Dot plots
location of the center (mean),
dispersion (range or standard deviation) and
shape (symmetry) of data
Outliers or unusual observations present?
Dotplot of Weight
Some Graphi
Arizona State University at the Polytechnic Campus
Elementary Statistics
MATHEMATIC 226

Summer 2016
7/31/2012
Variables and Data
Variables and Data
Two types of variables being measured
1. Qualitative Variables
2. Quantitative Variables
Qualitative variable measurements of this variable
are placed in one of several categories. Also called a
categorica
Arizona State University at the Polytechnic Campus
Elementary Statistics
MATHEMATIC 226

Summer 2016
8/10/2012
Distribution Shapes; Symmetry
and Skewness
Distribution Shapes; Symmetry
and Skewness
Definition:
The distribution of a set of data is a table, graph,
or formula that provides two things:
1. The values of the observations
2. How often those valu
Arizona State University at the Polytechnic Campus
Elementary Statistics
MATHEMATIC 226

Summer 2016
7/31/2012
Types of Statistics
Two types of statistics
1. Descriptive
Population and Sample
Definitions
Population
Methods for organizing and summarizing
information.
Graphs, charts, tables, calculation of averages,
standard deviations,
Sample
2. Infere
Arizona State University at the Polytechnic Campus
Linear Algebra
MATHEMATIC 343

Spring 2015
Page 1 of 7
MAT 343, SPR1NG2015, EXAM 1
FEBRUARY 19,2015 Name ( E
Technology is allowed for this test, but it should be used only for row reduction, matrix multiplication
and simple calculator functions. It should not be used for intemet access or stored