## October 31st 2013 Vector Homework

## The Venn Diagram of the History of Numbers

## Einstein Board

1. Gave time to finish the questions for the Addition of Vectors Lab with the group and had groups hand in the pink sheet with just the questions answered. Then went over how they needed each to place their own data table, conclusion, summary, heads of all vectors, etc. like this example.

2. Video for The Nightmare Before Christmas (3 min of the start)

3. Finished problems 8 & 9 on homework sheet Camera's, Action, ReAction (collected)

4. Polly want a Klacker time (Newton's 3rd Law in klackers)(note video below when I get it there this coming weekend.

5. Started problem 1 on the Unit 2 Problem Sheet

These are notes from last year 2012 you heard these after the last test in 2013.

Note in the diagram that humans started out working with just the (N)atural #'s (or counting numbers). The Natural #'s are 1, 2, 3, .... and BC most people did not know how to add, substract, multiply, and divide them but they did have a certain size pebble in a clay jar, one for each sheep in their flock so they could do a 1 to 1 correspondence with sheep (whatever item) and rocks to tell if one was missing.

Over a thousand years passed before the (W)hole number system was used which included zero. What is 5/0? No, it is not zero! You cannot divide by zero as you cannot have zero piles of something. Or in calculus if you approach zero from the right you get a + infinity and from the left a - infinity and since you do not know which side you are approaching zero from....it is easier to say that it cannot be done.

To jump to the chase for now....the big picture is that as you progressed into each new number system...you had to learn all over again how to simplify, add, subtract, multiply, and divide ... (and the identity, the inverse, and whatever other properties they have and don't have).

Vectors are no exception... You are entering a WHOLE NEW WORLD of VECTORS and they work differently then any of the previous number systems. You can add 1 plus 1 and get any number between 0 and 2 with vectors...depending on their directions.

You can always add vectors by moving a vector (keeping the same length (magnitude) and direction as to the rules) so that its tail hooks to the head of another...after hooking all the vectors tail to head the sum (resultant) is a new vector with its tail hooked to tail of the first vector in the string of vectors and its head is hooked to the head of the last vector in the string of added vectors.

Note in the diagram that humans started out working with just the (N)atural #'s (or counting numbers). The Natural #'s are 1, 2, 3, .... and BC most people did not know how to add, substract, multiply, and divide them but they did have a certain size pebble in a clay jar, one for each sheep in their flock so they could do a 1 to 1 correspondence with sheep (whatever item) and rocks to tell if one was missing.

Over a thousand years passed before the (W)hole number system was used which included zero. What is 5/0? No, it is not zero! You cannot divide by zero as you cannot have zero piles of something. Or in calculus if you approach zero from the right you get a + infinity and from the left a - infinity and since you do not know which side you are approaching zero from....it is easier to say that it cannot be done.

To jump to the chase for now....the big picture is that as you progressed into each new number system...you had to learn all over again how to simplify, add, subtract, multiply, and divide ... (and the identity, the inverse, and whatever other properties they have and don't have).

Vectors are no exception... You are entering a WHOLE NEW WORLD of VECTORS and they work differently then any of the previous number systems. You can add 1 plus 1 and get any number between 0 and 2 with vectors...depending on their directions.

You can always add vectors by moving a vector (keeping the same length (magnitude) and direction as to the rules) so that its tail hooks to the head of another...after hooking all the vectors tail to head the sum (resultant) is a new vector with its tail hooked to tail of the first vector in the string of vectors and its head is hooked to the head of the last vector in the string of added vectors.

## Vector Addition Homework Worksheet 2012

Period 1's progress with to scale drawings shown for problems 1 and 2

Today we start applying this knowledge of vector addition and their resultant to real life applications like with boats and current, and airplane flights with wind.

Today the class worked on problems 1 a, b; then 2 (period 1 above), then 1 c, d.

We then watched a demonstration of a sheet (representing the current) pulling a bulldozer traveling perpendicular to the direction of the moving sheet. This demonstration proved that the current is independent of the time it takes the boat (or swimmer or bulldozer) to cross the river. Thus to swim across a river (with or without a current) in the least amount of time...always swim perpendicular to the opposite shore.

Next each student had an etch-a-sketch that they drew the largest right triangle they could on. Note the big picture at the bottom of tomorrow (Resolution of a Vector Lab).

Today the class worked on problems 1 a, b; then 2 (period 1 above), then 1 c, d.

We then watched a demonstration of a sheet (representing the current) pulling a bulldozer traveling perpendicular to the direction of the moving sheet. This demonstration proved that the current is independent of the time it takes the boat (or swimmer or bulldozer) to cross the river. Thus to swim across a river (with or without a current) in the least amount of time...always swim perpendicular to the opposite shore.

Next each student had an etch-a-sketch that they drew the largest right triangle they could on. Note the big picture at the bottom of tomorrow (Resolution of a Vector Lab).

## Big Picture from 2012 but always relevant every year

1. Vectors are a Whole New World in the history of numbers to learn the rules and applications of. Only matrices are larger in the history of numbers. As you go up into the newer systems of numbers the applications become more and more powerful (useful), allowing you to accomplish more work in less time.

2. Traveling boats on currents are great applications for vector addition.

3. The Fab Five is now the Fab Ten as the direction now matters (it always did actually) so one uses the only horizontal vectors in a formula or one uses only vertical vectors in a formula. The two cannot be mixed. Example: To find the time it takes to cross the stream one uses the width of the stream and the vector in the same direction as the width. Also, to calculate the distance downstream...one uses only the vector representing the current velocity and the time. The time is both horizontal and vertical in nature.

4. A flying airplane and the wind makes for another great example of vector addition with a resultant.

2. Traveling boats on currents are great applications for vector addition.

3. The Fab Five is now the Fab Ten as the direction now matters (it always did actually) so one uses the only horizontal vectors in a formula or one uses only vertical vectors in a formula. The two cannot be mixed. Example: To find the time it takes to cross the stream one uses the width of the stream and the vector in the same direction as the width. Also, to calculate the distance downstream...one uses only the vector representing the current velocity and the time. The time is both horizontal and vertical in nature.

4. A flying airplane and the wind makes for another great example of vector addition with a resultant.