MindsOn#3.notebook
December08,2011
Minds On.
(i)
A river is 2 km wide and the current flows 6km/h.
James is driving a motor boat from one side of the
river to the marina directly across the river. The boat
is traveling at a speed of 20 km/h.
How long wil
MindsOn#7.notebook
December14,2011
Minds On .
Given the points A(3, 6, -1) and B(-1, 0 5), express AB
as a position vector.
Write a unit vector, u, in the direction of AB
Dec911:15AM
1
MindsOn#1.notebook
December07,2011
Minds On.
In his kayak, Jake paddles directly across a river at a
velocity of 10 km/h. At the same time the river is
flowing 6 km/h downstream.
What is the resultant speed of the kayak?
In which direction will the kaya
MatrixIntroduction.notebook
January17,2012
Matrices
We identify individual entries in the matrix with the
following notation:
amn
where m represents the row and n
represents the column where the entry is located.
A matrix is an ordered set of numbers list
LinearCombiations&SpanningSets.notebook
Linear Combinations & Spanning Sets
December09,2011
If a = (3, -2) and b = (2, 4), what combination of a
and b will create (12, 8)?
Algebraic vectors, written in unit vector form are
linear combinations.
OP = (4, -
LimitsIntro.notebook
January21,2012
Limits of a Fcn
MEANS:
"limit of the function f(x) as x
approaches 'a' is a number L"
A limit will only exist for a function
as x approaches 'a' if:
"limit of the function f(x) as x
approaches 'a' from the right is a
nu
IntersectingPlanes.notebook
January14,2012
Determining Intersection Of Planes
(i)
1: 2x - y + z - 1 = 0
2: x + y + z - 6 = 0
DotheyIntersect?How?
Intersection of Two Planes
How Can Planes Intersect?
#1
#2
Jan1311:01AM
DotheyIntersect?How?
(ii) 1: 2x - 6y
Intersectionof3Planes.notebook
January19,2012
Intersection of Three Planes
Three Planes can intersect:
#1. Consistent
Determine if the following planes intersect. If they
do, determine the point of intersection. If they don't,
describe their characteristi
IntersectionLines&Planes.notebook
January12,2012
How Can We Determine Intersections of Planes &
Lines?
Intersection:
A Line with a Plane & Two Lines
(i)
l1: r = (1, 2, -1) + (2, -4, 7)t
: 4x -5y -4z + 2 = 0
Do they intersect? Where?
How Can a Line inte
The Fish Caught in the Sea
To make the plotting of the data points easier and more convenient, the year was assumed to start
from 0 and increase by 1, therefore (1980 is 0), (1981 is 1), and (2006 is 28). After the Year
versus Total Mass was plotted on a
EquationsofLinesinR3.notebook
Vector, Parametric & Symmetric Equations in R3
We can derive the equation of a line in R3 using the
techniques discussed with the equations of lines in
R2.
If we are given two points on the line we must
determine the directi
DotProductII.notebook
December15,2011
Dot Product II - Algebraic Vectors
x
B(b1,b2)
Dot Product
A(a1,a2)
O(0,0)
x
Ex: Find the dot product for the given vectors.
(i) a = (4, 2)
b = (1, 3)
(ii) a = (4, 3, 1)
b = (2, -1, 4)
Dec92:22PM
Dec92:34PM
Determine t
DotProductI.notebook
Dot Product
We have discussed:
Vector Addition & Subtraction
Scalar Multiplication of Vectors
December10,2011
Max is pulling his sled up a hill with a force of 120 N at an
angle of 20o to the surface of the hill. The hill is 100m l
DerivativeFirstPrinciples.notebook
January21,2012
Introduction to Derivatives
The concepts of limits and rates of change are the
foundation of an operation called Differentiation.
This is a key concept in the study of Calculus.
Derivative to a function ma
Culminating Activity for Unit 1
Select three jobs: one knowledge-based, one manufacturing, and one in the service
sector. Based on your search of the websites of Human Resource Development Canada,
Stats Canada, and other relevant Web sources, what pattern
CrossProduct.notebook
December15,2011
Cross Product
The cross product of two vectors, a x b, creates a
vector quantity.
The vector product is the vector perpendicular to
both vectors a and b.
Cross product will only work for non-collinear and non
zero
CartesianEqnofaPlane.notebook
January11,2012
Cartesian Equation of a Plane
Cartesian equation of a line - require a
point and the vector perpendicular to the
direction vector - normal vector, n.
Cartesian equation of a plane - requires a
point and a vec
CartesianEqnofLine.notebook
December22,2011
Slope & Direction Vectors
Cartesian Equation of a Line in R2
How are the equations y = mx + b & Ax + By + C = 0
related to the vector & parametric equations of a line?
Thedirectionvectorhasatail@(0,0)andatipat
D
VectorIntroduction.notebook
December08,2011
Introduction to Vectors
Suppose you are flying from Kingston to
Toronto. Draw a flight path for this flight.
170
180
Introduction to Vectors
160
150
20
140
0
10
30
130
40
50
120
60
Scalar - a quantity that is fu
Vector&ParametricEqnofLinesinR2.notebook
December21,2011
Vector & Parametric Equations of a
Line in R2
y
(1,4)
x
*Consider the line with
a slope of 2/3 and
passing throught the
point A(-1,4).
* What is the equation
of the line?
Lines can be drawn in R3 as
Vector&ParametricEqnofPlanes.notebook
January10,2012
Equations of a Plane
A PLANE is a two-dimensional representation in R3.
A plane is like a table top or a wall that extends indefinitely
in two directions.
A plane is drawn as a parallelogram on a XYZ
TrigOptimization.notebook
November26,2011
Optimization
Shed, 3m high and 2m from front to back, is
located beside a wall. Find the length of the
shortest ladder that can reach up the wall. What
angle will the ladder make with the ground? Wall?
Automotive
RelatedRatesTrig.notebook
January21,2012
Related Rate using Trigonometry
A lighthouse beacon is 500 m from
shore. The light makes one
revolution every 45 seconds. How
fast is the spot on the shore moving
along the shore when the spot is
250m from the clos
Optimization3.notebook
January21,2012
More Optimization
An electronics store is selling MP3 players. The
regular price for the MP3 player is $90. During
a typical month, the store sells 100 units. Past
sales have shown that for every $1 decrease in
price,
Optimization2.notebook
January21,2012
Minimizing Costs
I am building a new cottage on an island that is 400m from the
nearest point on shore. I need to run a power line to the
island, but the power line ends 3km from the nearest point on
shore. The cost t
Optimization1.notebook
January21,2012
The Biggest Box?
Criteria:
use paper 20cm x 25cm
open top box
rectangular base
uniform height
20 cm
25 cm
Oct1311:51AM
Oct1311:52AM
Using Calculus:
20 cm
By Investigating:
Length(l)
Width(w)
Height(h)
Volume(V)
25
OperationswithVectorsII.notebook
December08,2011
Operations with Vectors II
Scalar Multiplication
For the vector ka, where k is a scalar and a is a
non-zero vector, then
Collinear Vectors - vectors are colliniear if they are parallel - they
can be transla
OperationswithVectorsI.notebook
December08,2011
Operations with Vectors I
There are no direct flights from Kingston to Syracuse. All flights
must pass through Toronto first.
The diagram from the flight path represents the addition of two vectors
using the