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Employ an online or offline tutor at your own risk from AU: tutorfinder.com.au CDN : findatutor.ca CDN: .i-tutor.ca CDN: Montreal Tutors NZ: findatutor.co.nz UK: tutorhunt.com UK: tutors4me.co.uk USA: wiziq.com USA: ziizoo.com |
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YOU are better than YOU think. Show yourself how: |
-/[]\- Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck. |
-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
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Explore collaborative whiteboards from groupboard, twiddla or scriblink. |
1 Arrows or Vectors - Directed Line Segments in the Plane
We need the concept of a directed line segment along a straight line and in the plane. A directed line segment in the plane or in the line will also be called an arrow or vector.
Geometric View of Arrows In the Plane
(Navigation)
On a map, a sequence of straight line motions may be used to precisely or approximately represent the path of an object (ship, plane or person) over land or sea. These motions and their directions may be represented by arrows with tail at the starting point of a motion and head at the other end or last point in that motion. Here is Motivation and a context for the use of arrows, or vectors, in navigation.
Directed line segments initial points and terminal points, or arrows or vectors with heads and tails may be used to describe or show straight line movements and the direction of motion in the plane or along a line.
In the next figure, the path of the sailboat takes it from A to B, then B to D, then D to G, then G to C, then C to H and then H to M. Think of this as the head-to-tail map addition of movement or vectors.
2. Addition - Displacement, Head to Tail in the Plane.
Resultant of Movements.
A straight line arrow from one point to another may summarize the movement of an object. The object itself may follow a curved path between the tail or initial point of the arrow and the head or terminal point. Similarly when a sequence of straight line motions is followed, one after another, the arrow joining the initial point of the first motion to the terminal point of the last motion summarizes or gives the sum or resultant of the intermediate motions. Here is a context and motivation for the head to tail addition of arrows or vectors in navigation.
The straight line path between the start and finish of a sequence of movements (shown as arrows on maps) is called their resultant or sum. The vector from A to M gives an example of a resultant, the net result of a sequence of motions. In this map-based example, the vector from A to M represents the top view of a path a bird could fly if does not represent a possible path of the boat. It could be the path if the land was not there -- raise the water level.
Addition of Arrows: The directed line segment from the initial point A to the head point M gives what we call the Sum of Arrows or Directed Line Segments A to B, B to D, D to G, G to C, C to H and H to M.
3. Associativity of In-Place Head-to-Tail Addition
Similar diagrams should imply associatively for in-place head-to-tail addition of any three vectors in the plane or in a line.
Assumption: In place head-to-tail addition of vectors is associative.
www.whyslopes.com
Number Theory
Start of Number Theory
Origins of Counting or Tallying
Number Theory
Continued
Related Site Pages:
For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck
Food for thought: Try the Twiddla Whiteboard. In principle, it allows to people to draw and chat together online on a copy of this webpage or a clean sheet. The chat may be via text or audio. Visit www.twiddla.com to set up whiteboards to work with the webpage of your choice..