Title: Points in Space

Title: Points in Space

IN THE SWISS COUNTRYSIDE

SOME IMPORTANT SCIENTIFIC WORK IS TAKING PLACE.

SO AS NOT TO OBSTRUCT THE VIEW OF THE ALPS,

THIS WORK IS HAPPENING UNDERGROUND.

CERN, THE EUROPEAN AGENCY

THAT DOES RESEARCH IN SUB ATOMIC PHYSICS

HAS RECENTLY LAUNCHED THE LARGE HADRON COLLIDER.

THIS CIRCULAR TUNNEL WILL ACCELERATE

SUBATOMIC PARTICLES TO NEARLY THE SPEED OF LIGHT

AND HAVE THEM COLLIDE INTO EACH OTHER.

TO UNDERSTAND WHAT A SUBATOMIC PARTICLE IS

LET'S START WITH AN ATOM.

AN ATOM IS ONE OF THE SMALLEST PARTICLES

THAT CAN STILL BE CALLED A SUBSTANCE.

GOLD CAN EXIST AS AN ATOM

BUT THERE IS NO SUBATOMIC VERSION OF GOLD.

AN ATOM INCLUDES ELECTRONS, NEUTRONS AND PROTONS.

LET'S LOOK AT THE SIMPLEST ATOM, HYDROGEN,

WHICH CONSISTS OF A PROTON AND AN ELECTRON.

THE PROTON IS A SUBATOMIC PARTICLE WHOSE SIZE

IS IN THE NEIGHBORHOOD OF 10 TO THE -13TH METERS,

WHICH IS INFINITESIMALLY SMALL.

IN TERMS OF GEOMETRY, IS IT POSSIBLE TO

CONSIDER A PROTON A GEOMETRIC POINT?

AFTER ALL, IT IS SO SMALL THAT IT CAN'T BE SEEN

BY THE HUMAN EYE.

WHAT WE NEED IS A MATHEMATICAL DEFINITION

OF A GEOMETRIC POINT.

WE CAN USE EUCLID'S DEFINITION.

WHAT THIS MEANS IS THAT A POINT HAS NO SIZE

OR DIMENSION.

HOWEVER SMALL A SUBATOMIC PARTICLE IS,

IT IS STILL A MEASUREMENT IN SPACE.

A GEOMETRIC POINT HAS NO SIZE

BUT SIMPLY A LOCATION IN SPACE.

WE CAN USE THIS CONCEPT OF A LOCATION IN SPACE

TO HELP EXPLAIN WHAT HAPPENS WITH SUBATOMIC PARTICLES.

HERE ARE TWO POINTS LABELED "A" AND "B".

SUPPOSE THAT EACH REPRESENTS THE LOCATION

OF A SUBATOMIC PARTICLE IN THE LARGE HADRON COLLIDER

AND SUPPOSE THESE PARTICLES ARE MOVING TOWARD EACH OTHER.

THERE ARE THREE POSSIBLE OUTCOMES.

IN ONE OUTCOME, A AND B MOVE PAST EACH OTHER

WITH A MOVING ABOVE B.

IN ANOTHER OUTCOME, A AND B MOVE PAST EACH OTHER

WITH A MOVING BELOW B.

IN THE THIRD SCENARIO, THE ONE WE'LL BE EXPLORING,

A AND B COLLIDE.

IN THE CASE OF THE SUBATOMIC PARTICLES,

WHEN THEY COLLIDE SOMETIMES SPARKS FLY.

LET'S SEE WHAT THIS MEANS GEOMETRICALLY

BY USING THE TI-NSPIRE.

TURN ON THE TI-NSPIRE.

CREATE A NEW DOCUMENT.

YOU MAY NEED TO SAVE A PREVIOUS DOCUMENT.

CREATE A GRAPHS AND GEOMETRY WINDOW.

SELECT THE POINT TOOL.

CLICK ON MENU, AND UNDER "POINTS AND LINES"

SELECT POINT.

MOVE THE POINTER TO THE MIDDLE PART OF THE SCREEN

AND PRESS CLICK TO CREATE A POINT.

MOVE THE POINTER TO A DIFFERENT PART OF THE SCREEN

AND CREATE A SECOND POINT.

NOW LABEL EACH POINT.

PRESS ESCAPE AND MOVE THE CURSOR

ABOVE THE FIRST POINT.

PRESS AND HOLD THE CLICK KEY.

PRESS CONTROL AND MENU AND SELECT THE LABEL OPTION.

PRESS THE CAPS KEY AND THE LETTER A TO LABEL THE POINT.

REPEAT THE LABELING PROCESS WITH THE OTHER POINT.

LABEL IT B THEN PRESS ESCAPE.

MOVE THE CURSOR OVER POINT A.

PRESS AND HOLD THE CLICK KEY TO GRAB THE POINT.

MOVE THE POINT SO THAT IT OVERLAPS POINT B.

WHEN YOU PLACE ONE POINT OVER ANOTHER THIS WAY,

WHAT IS HAPPENING GEOMETRICALLY?

THE POINTS INTERSECT AND SHARE THE SAME LOCATION

IN SPACE.

REMEMBER THAT POINTS HAVE NO DIMENSION

SO IT'S NOT AS IF ONE POINT IS CROWDING OUT ANOTHER.

POINTS ARE NOT LIKE PARTICLES IN SPACE.

IF THE POINTS SHARE THE SAME POSITION IN SPACE

THEN THE DISTANCE BETWEEN POINTS A AND B IS ZERO.

NOW POSITION THE POINTS SO THAT

THERE IS SOME DISTANCE BETWEEN THEM.

LET A REPRESENT A SUBATOMIC PARTICLE

MOVING IN THE DIRECTION OF B AND LET B REPRESENT

A SUBATOMIC PARTICLE MOVING IN THE DIRECTION OF A.

SINCE THE DISTANCE BETWEEN A AND B IS GREATER THAN ZERO,

THEN THERE IS A THIRD POINT, C, MIDWAY BETWEEN A AND B.

OTHERWISE THE DISTANCE BETWEEN A AND B

WOULD BE ZERO.

CONTINUING WITH THIS, THERE IS A POINT BETWEEN A AND C

AND BETWEEN C AND D MIDWAY BETWEEN THOSE POINTS.

THIS PROCESS CAN CONTINUE INFINITELY.

WHY?

BECAUSE A POINT REPRESENTS A POSITION IN SPACE,

NOT AN AMOUNT OF SPACE.

THERE ARE AN INFINITE NUMBER OF POINTS BETWEEN A AND B.

THIS INFINITE NUMBER OF POINTS BETWEEN A AND B

REPRESENTING THE PATH THAT THE TWOSUBATOMIC PARTICLES

HAVE THAT COLLIDE WITH EACH OTHER

GIVE RISE TO ANOTHER GEOMETRIC FORM: THE LINE.

WE CAN USE EUCLID'S DEFINITION OF A LINE.

A LINE IS BREADTHLESS LENGTH.

WHAT THIS MEANS IS THAT A LINE HAS A LENGTH

THAT CAN BE MEASURED BUT NOT A WIDTH OR HEIGHT.

BECAUSE IT IS MADE UP OF DIMENSIONLESS POINTS

IT BECOMES A ONE DIMENSIONAL FIGURE.

SO TWO SUBATOMIC PARTICLES THAT COLLIDE WITH EACH OTHER

ARE MODELED BY TWO COLLINEAR POINTS,

OR POINTS THAT ARE ON THE SAME LINE.

TO CONSTRUCT A LINE, PRESS MENU

AND UNDER "POINTS AND LINES", SELECT LINE.

MOVE THE CURSOR ABOVE POINT A

UNTIL THE POINTER TURNS INTO A POINTING HAND.

PRESS ENTER.

THEN MOVE THE POINTER TOWARDS POINT B.

NOTICE HOW A LINE FOLLOWS THE MOVEMENT OF THE POINTER.

WHEN THE POINTER IS ABOVE B PRESS ENTER AGAIN.

YOU NOW HAVE THE LINE CONNECTING POINTS A AND B.

IS THERE MORE THAN ONE LINE THAT CAN INTERSECT

BOTH POINTS A AND B?

USE THE LINE TOOL TO CONSTRUCT OTHER LINES

THAT CONTAIN AT LEAST ONE OF THE POINTS.

YOU CAN EASILY CREATE MANY LINES

THAT CONTAIN ONE OF THE POINTS.

YOU CAN EVEN CONSTRUCT LINES

THAT HAVE NONE OF THE POINTS.

BUT THERE IS ONLY ONE LINE THAT CROSSES THE TWO POINTS.

WE CAN GENERALIZE THIS TO SAY THAT FOR ANY TWO POINTS

THERE IS A UNIQUE LINE THAT CROSSES THE TWO POINTS.

IN OTHER WORDS, ANY TWO POINTS ARE COLLINEAR.

CONSTRUCT A NEW LINE.

FIND A CLEAR PART OF THE SCREEN.

PRESS THE CLICK KEY ONCE.

THEN MOVE THE POINTER TO ANOTHER PART OF THE SCREEN.

PRESS THE CLICK KEY AGAIN TO COMPLETE THE LINE.

NOTICE THAT YOU ARE IN "POINT ON" MODE.

IN THIS MODE ANY POINTS THAT YOU ADD TO THE LINE

ARE COLLINEAR.

ADD SEVERAL MORE COLLINEAR POINTS.

NOW ADD A POINT THAT ISN'T ON THE LINE.

THINK OF THE SITUATION OF THREE POINTS.

HOW CAN YOU ENSURE THAT THEY ARE COLLINEAR?

CLEAR YOUR GEOMETRY WINDOW.

PRESS MENU AND UNDER ACTIONS SELECT "DELETE ALL".

CLICK OKAY TO CONFIRM THE DELETION.

SELECT THE POINT TOOL. PRESS MENU,

AND UNDER "POINTS AND LINES" SELECT POINT.

PLACE THREE POINTS ON SCREEN THAT ARE CLEARLY

NOT COLLINEAR.

YOU KNOW THAT FOR ANY TWO POINTS THERE IS A

UNIQUE LINE THAT INCLUDES THE TWO POINTS.

BUT WHAT IS THE UNIQUE LINE THAT CAN INCLUDE ALL THREE?

SINCE THE THREE POINTS ARE NOT COLLINEAR THIS ISN'T

POSSIBLE, BUT THERE IS A WAY OF MAKING THEM COLLINEAR.

ACCESS THE LINE TOOL.

CLICK ON ONE OF THE THREE POINTS AND CREATE

A LINE TO ANOTHER ONE OF THE POINTS.

THE TWO COLLINEAR POINTS ARE ON THE LINE

AND THE THIRD POINT IS NOT ON THE LINE.

TO MAKE THE THREE POINTS COLLINEAR

HIGHLIGHT THE THIRD POINT AND PLACE IT ON THE LINE.

A STREAM OF SUBATOMIC PARTICLES ARE MOVING

TOWARD EACH OTHER.

THOSE THAT COLLIDE WITH EACH OTHER

CAN BE MODELED BY COLLINEAR POINTS.

OR THEY CAN BE MODELED BY INTERSECTING LINES.

TAKE A LOOK AT POINTS A AND B.

WE KNOW THAT THERE IS A UNIQUE LINE

THAT INCLUDES BOTH POINTS.

BUT THERE ARE ALSO AN INFINITE NUMBER OF LINES

THAT INCLUDE ONE OF THE POINTS.

CLEAR YOUR GEOMETRY WINDOW.

PRESS MENU AND UNDER ACTIONS SELECT "DELETE ALL".

CLICK OKAY TO CONFIRM THE DELETION.

SELECT THE POINT TOOL. PRESS MENU,

AND UNDER "POINTS AND LINES" SELECT POINT.

PLACE TWO POINTS ON SCREEN.

USE THE LINE TOOL TO CREATE TWO LINES IN SUCH A WAY

THAT EACH LINE HAS ONE OF THE TWO POINTS.

MAKE SURE THAT THE LINES INTERSECT.

ACTIVATE THE LINE TOOL.

MOVE THE POINTER TO WHERE THE LINES INTERSECT.

NOTICE THE ON-SCREEN MESSAGE THAT SAYS

"INTERSECTION POINT". PRESS ENTER.

YOU NOW HAVE A THIRD POINT WHERE THE LINES INTERSECT.

THE THREE POINTS ARE NOT COLLINEAR SINCE

THERE ISN'T ONE LINE THAT CONTAINS ALL THE POINTS.

BECAUSE A GEOMETRIC POINT IS A LOCATION IN SPACE,

THIS PROPERTY ALLOWS US TO MODEL

THE LOCATIONS OF SUB ATOMIC PARTICLES.

WHEN YOU KNOW THE LOCATION OF AN OBJECT

YOU CAN USE COORDINATES TO TRACK POSITION.

ASSIGNING COORDINATES TO POINTS

IS AN EXAMPLE OF COORDINATE GEOMETRY.

LET'S LOOK AT AN EXAMPLE ON THE INSPIRE.

PRESS THE HOME KEY AND CREATE A NEW

GRAPHS AND GEOMETRY WINDOW.

NOTICE THE COORDINATE GRID THAT APPEARS.

DRAW TWO POINTS ON THE GRID. PRESS MENU,

AND UNDER "POINTS AND LINES" SELECT POINT.

MOVE THE POINTER TO THE MIDDLE OF THE SCREEN

AND PRESS ENTER TO CREATE THE FIRST POINT.

MOVE THE POINTER TO A DIFFERENT PART

OF THE SCREEN. PRESS ENTER AGAIN.

YOU NOW HAVE TWO POINTS.

EACH OF THESE POINTS HAS A POSITION IN SPACE

AND HAS COORDINATES ASSIGNED TO THEM

IN THIS COORDINATE SYSTEM.

TO SEE THE COORDINATES PRESS MENU AND UNDER ACTION

SELECT "COORDINATES AND EQUATIONS".

USE THE NAV PAD TO MOVE THE POINTER

ABOVE ONE OF THE POINTS.

YOU'LL SEE COORDINATES APPEAR NEXT TO THE POINT.

PRESS ENTER TWICE TO PASTE THE COORDINATES ON SCREEN.

REPEAT WITH THE OTHER POINT.

YOU CAN ALSO ASSIGN DIFFERENT COORDINATES

TO THESE POINTS.

WHEN YOU DO SO, THE POINTS WILL MOVE

TO THE APPROPRIATE LOCATIONS.

PRESS THE ESCAPE BUTTON.

NOW MOVE THE POINTER ABOVE THE X COORDINATE

OF ONE OF THE POINTS.

PRESS ENTER TWICE TO SEE A CURSOR APPEAR

WITHIN THE TEXT FLD OF THE COORDINATE.

PRESS THE CLEAR BUTTON SEVERAL TIMES

TO DELETE THE COORDINATE.

REPLACE THE COORDINATE WITH THE NUMBER 2.

PRESS ENTER.

NOTICE THAT THE POINT SHIFTS HORIZONTALLY

SINCE YOU CHANGED THE X COORDINATE.

REPEAT THIS PROCESS FOR THE REMAINING COORDINATES.

CHANGE THE Y COORDINATE OF THE FIRST POINT TO 3

AND CHANGE THE X,Y COORDINATES

OF THE SECOND POINT TO -2, -4.

TRY TO GET YOUR SCREEN TO LOOK LIKE THIS.

GEOMETRIC POINTS DO NOT HAVE SIZE.

THEY ARE LOCATIONS IN SPACE.

GEOMETRIC SPACE IS MADE UP OF ALL GEOMETRIC POINTS.

BECAUSE A POINT HAS A LOCATION,

COORDINATES CAN BE ASSIGNED TO IT.

THE MOVEMENT OF AN OBJECT IN SPACE CAN BE THOUGHT OF

AS THE CHANGE IN LOCATION OF A SET OF GEOMETRIC POINTS.