# 4.1 Represent, Analyse & Calculate Shape & Motion In 2- & 3 Dimensional Space In Different Contexts

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**1 Measure, Estimate, & Calculate Physical Quantities In Practical Situations Relevant To The Adult**

**Assessment criteria:**

We measure to find out about an object. A carpenter needs to know the length of a piece of wood. It has to be the right size. People who work in the city need to know how long the train journey will take to get to work.

When using a measuring instrument, ensure that the measurement is:

- Accurate
- Precise
- Viewed squarely off the scale of the measuring instrument

**Outcomes range:**

*Basic instruments:*

**Rulers**

A ruler that is in a good condition is a practical instrument for measuring shorter, straight lines. We can measure millimetres and centimetres with a ruler. If you need to measure using a ruler, you will place the 0 and read the measurement at the end of the line on the comparative point on the ruler.

**Measuring tapes**

A measuring tape is used when a ruler is too short to measure the distance or length. We use the measuring tape to measure short distances in metres.

**Measuring cylinders and jugs**

When we need to measure the capacity of a container to determine how many millilitres or litres are in the container, we use measuring cylinders or jugs. Theses cylinders or jugs have markings that show the measurement if you want to take the measurement of liquid, the level does not form a straight line, but a line that is curved at the edges.

You need to:

- Stand level with the line
- Ignore the curved edges.
- Look squarely at the level.
- Take the measurement where the straight line forms.

**Thermometers**

We use a thermometer to measure temperature. Thermometers are often filled with mercury or alcohol. The higher the temperature rises, the more the mercury or alcohol expands to show the temperature reading on the scale.

**Kitchen balances/Scales**

When we need to find the mass of an item, then we use a spring scale or a kitchen scale. We measure the mass of something in grams or kilograms.

**Watches and clocks**

We can measure time either on an analogue or digital clock. The analogue clock is marked in 12 hour intervals and the clocks hands need to pass the 12 for the second time to indicate a 24 hour period. A digital clock works on a 24 hour basis. We do not use a.m. or p.m. to indicate the time of day, because on a digital clock 5:00 a.m. is 05h00 and 5:00p.m. is 17h00.

*Workplace instruments:*

__Vernier callipers__: a device for measuring widths or distances, consisting of two thin long moveable pieces of metal fixed together at one end.

__Micrometer screws__: A device used for making very exact measurements or for measuring very small things.

__Stop watches__: A watch that can be started or stopped in order to measure the exact time of an event, eg. a sports event.

__Chemical balances__: A device used for weighing things. It consists of two dishes hanging on a bar that shows when the contents of both dishes weigh the same.

*Quantities:*

__Length/distance__: The measurement of something from end to end or along its longest side.

__Area: __The measure of a flat space.

__Mass:__ A large amount of something that has no particular shape or arrangement.

__Mass – matter;__ Specialised the amount of matter in any solid object or in any volume of liquid or gas.

Time: Part of existence which is measured in seconds, minutes, hours, days, weeks, months, years, etc. or this process considered as a whole.

Speed acceleration: To move more quickly.

__Temperature:__ The measured amount of heat in a place or in the body.

*Distinctions between :*

Mass and weight: Weight – to measure the heaviness of an object. Mass – the amount of matter in any solid object.

Speed and acceleration: Speed – Rate of movement. Acceleration – to move more quickly, or to make (something) happen faster or sooner.

**Quantities**:

Low or small

High or large

Calculate heights and distances using the Pythagoras theorem.

**The Pythagoras theorem:**

An explanation of the theorem from the website -askjeeves.com

**The Pythagorean Theorem**

by

Stephanie J. Morris

*The Pythagorean Theorem was one of the earliest theorems known to ancient civilizations. This famous theorem is named for the Greek mathematician and philosopher, Pythagoras. Pythagoras founded the Pythagorean School of Mathematics in Cortona, a Greek seaport in Southern Italy. He is credited with many contributions to mathematics although some of them may have actually been the work of his students.*

*The Pythagorean Theorem is Pythagoras’ most famous mathematical contribution. According to legend, Pythagoras was so happy when he discovered the theorem that he offered a sacrifice of oxen. The later discovery that the square root of 2 is irrational and therefore, cannot be expressed as a ratio of two integers, greatly troubled Pythagoras and his followers. They were devout in their belief that any two lengths were integral multiples of some unit length. Many attempts were made to suppress the knowledge that the square root of 2 is irrational. It is even said that the man who divulged the secret was drowned at sea.*

The Pythagorean Theorem is a statement about triangles containing a right angle. The Pythagorean Theorem states that:

*“The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.”*

**Figure 1**

According to the Pythagorean Theorem, the sum of the areas of the two red squares, squares A and B, is equal to the area of the blue square, square C.

Area Square A = a²

Area Square B = b²

Area Square C = c²

Thus, the Pythagorean Theorem stated algebraically is:

a² + b² = c²

for a right triangle with sides of lengths a, b, and c, where c is the length of the hypotenuse Although Pythagoras is credited with the famous theorem, it is likely that the Babylonians knew the result for certain specific triangles at least a millennium earlier than Pythagoras. It is not known how the Greeks originally demonstrated the proof of the Pythagorean Theorem.

**1.1 Scales On The Measuring Instruments Are Read Correctly**

Activity 1:Do the following readings on measuring instruments in your workbooks.

- Ruler
- Measuring tape.
- Measuring Cylinder.
- Thermometer
- Kitchen balances.
- Watches and clocks.
(A practical session on reading of measuring instruments will be done in group format.)

**1.2 Quantities Are Estimated To A Tolerance Justified In The Context Of The Need**

Estimation is the technique of assigning approximate values to numbers in calculations, often in order to make calculation faster and easier. Estimation is valuable where precision is unnecessary, or where working with pencil and paper or calculator is not feasible, or to provide a double check on a precise calculation. In science it may be used as the first test of a theory to see if it is reasonable. It is of particular relevance in astronomy, a field in which many measurements cannot be taken accurately. Estimation is not appropriate where the final result needs to be precisely accurate, as in many scientific contexts and in formal financial transactions.

For example, when we want to estimate the size of a room, we will not necessarily measure it with a measuring tape, but take large steps that are roughly equal to a metre. In this case we are estimating the size of the room. However when we need to work out the exact answer, then we will measure the exact space with a suitable measuring instrument.

Activity 2:Estimate the size of the training room. Describe your chosen method of estimation. Now measure the exact size of the training room and check the accuracy of your estimation.

**1.3 The Appropriate Instrument Is Chosen To Measure A Particular Quantity**

Activity 3:Complete the table in you workbooks to indicate the instrument chosen for the particular quantity.

**1.4 ****Quantities Are Measured Correctly To Within The Least Step Of The Instrument **

Activity 4:Choose 5 measuring instruments and indicate the least step of the instrument.## Example: Ruler – Millimetres.

**1.5 Appropriate Formulae Are Selected & Used**

** **See AC1.6

**1.6 Calculations Are Carried Out Correctly & The Least Steps Of Instruments Used Are Taken Into Account When Reporting Final Values**

* **The information below will cover assessment criteria 1.5 and 1.6.*

*Length and breadth:*

Length is always the longer side of a shape, while breadth is the shorter side. In the examples below, the double lines indicate the length while the single lines indicate the breadth. Length and breath are indicated in metres. Any shape that has length and breadth is a two-dimensional shape**. **

*Perimeter and circumference:*

Perimeter is the distance from one point on the outside border of a shape, all the way around, to the same point again. Perimeter and circumference are measured in metres.

To calculate the perimeter of a rectangle, square or parallelogram:

P= 2 lengths + 2 Breadths

And the answer is in mm,cm,m or km.

Example:

To calculate the perimeter of the figure below:

To calculate the perimeter of any straight-lined shape with more than four angles: P = total sum of the length of all the straight lines.

** **

*Circumference*

Circumference is similar to perimeter, but circumference is the word we use to describe the ’perimeter’ of a circular shape.

** **

** **** Point A**

** **To calculate the circumference of a circle:

C = 2pr

Where p is 22/7 or 3,14

Example:

To calculate the circumference of the circle below, where p = 3,14:

** **

** **

*Area and volume*

Area is the amount of space a shape takes up in two dimensions, i.e. length and breadth.

Volume is the space that a container can take on the inside. In order to determine the volume of a container, we need to add another dimension to the shape namely the height or depth**.**

** **

Activity 5:Do the following calculations:

- The area of a rectangle 5cm x 4cm.
- The area of a triangle, height 5cm and breadth 7cm
- The area of a square, height 15cm and breadth 15cm
- The volume of a cube 7cm length, 7cm length,7cm length

**1.7 Symbols & Units Are Used In Accordance With SI Conventions & As Appropriate To The Situation.**

**The International System of Units (SI)**

The System International (SI) has common units to measure quantities that make communicating measurement between countries easy. The SI unit system is based on the metric system, which means that we use 10 (and its subsequent multiples) as the base number to increment various measures.

**Meter**for distance. Length is the distance between two points**Kilogram**for mass: Mass is the measurement for the quantity of substance present in the body. Mass is different from weight. Weight is the measure of attraction of the earth for a given mass and is measured in Newton**Seconds**for time: Time is what we use to measure the period that an action or event takes to occur. We measure time in hours, minutes and seconds.**Litre**for capacity: capacity refers to the amount of substance (usually a liquid) that is contained in a specific space. e.g. When we buy a 2 litre Cola, we expect it to contain 2 litres of cool drink.**Degrees Celsius**for temperature: Temperature is the condition or degree of warmness or coldness of a body. The weather report shows us the maximum and minimum temperature of the next 24 hour period. We usually use a thermometer to measure temperature. In South Africa we use degrees Celsius and in some other countries, degrees Fahrenheit is the standard measure for temperature.**Square metre**for area: Area is the size that a plane surface or two-dimensional shape takes up between its boundary lines (length and breadth). Area is calculated by multiplying the length with the breadth.**Cubic meter**for volume.**Velocity in metre per second**: velocity is the rate of motion or speed that something travels in a particular direction.**Metre per second squared**for acceleration: Acceleration is the rate at which the velocity of an object is increased per unit of time.**Hertz**for frequency: Frequency is the rate at which something occurs in one second.**Newton**for force: Force is the influence that is used so that change or movement takes place. Force can make things move, bring something to rest or make objects change direction.

Activity 6:Make a list of the symbols and SI conventions that you will use in your venture. Give an example of where/how you will use these conventions. What do you understand under ”SI” ?

** **

**2 Explore, Analyse & Critique, Describe & Represent, Interpret & Justify Geometrical Relationships**

- Applications taken from different contexts such as packaging. arts, building construction, dressmaking.
- The operation of simple linkages such as car jacks.
- Top, front and side views of objects are represented.
- Use rough sketches to interpret, represent and describe situations.
- The use of available technology (e.g. isometric paper, drawing instruments, software) to represent objects.
- Use and interpret scale drawings of plans.
- Road maps relevant to the country.
- World maps.
- International time zones.
- The use of the Cartesian co-ordinate system in determining location and describing relationships in at least two dimensions.

**2.1 Descriptions Are Based On A Systematic Analysis Of The Shapes & Reflect The Properties Of The Shapes Accurately, Clearly & Completely.**

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**2.2 Descriptions Include Quantitative Information Appropriate To The Situation & Need.**

**Rectangle **

**Square**

**Parallelogram**

** **

**Rh****ombus**

A rhombus is a quad of which all four sides are equal in length but none of the interior angles are equal to a right angle.** **

AB = BD = CD =AC

AB//CD and BD//AC

**Trapezium**

A trapezium is a quad with one pair of opposite sides parallel to each other but not necessarily equal in length.

AB//DC

** **

**Kite **

A kite is a quad that has two pairs of adjacent sides equal in length and opposite sides not equal in length.

AB = AD and BC

Activity 7:Study the following shapes and make a sketch of each in your workbooks. Write the main characteristic of each shape next to your drawing.

- Scalene triangle
- Isosceles triangle
- Equilateral triangle
- Right – angled triangle

**2.3 3-Dimensional Objects Are Represented By Top, Front & Side Views. **

A two-dimensional figure is a figure that has only two dimensions, namely length and breadth.

**Breadth**

A three-dimensional figure is a figure that has three dimensions, namely length, breadth and height or depth.

Activity 8:Make rough sketches in your workbooks of the following article. View these articles from the top, the side and the front.

- Motorcar
- Chair
- Tin of condensed milk
- A book
Complete the following activity in your workbooks.

- Draw the following two-dimensional shapes and transform them to three dimensional shapes by adding a height or depth of 3 cm.

- Square with dimensions 250mm.
- Rectangle with dimensions 300mm by 200mm.
- Circle with a diameter of 500mm.

**2.4 Different Views Are Correctly Assimilated To Describe 3-Dimensional Objects.**

Activity 8:Practical exercise: Redraw the objects displayed in the class as if you are looking from the top.

**2.5 Available & Appropriate Technology Is Used In producing & Analysing Representations**

*You’ve got to see it to believe it.*

–Anonymous

Did you know that most humans absorb more than 80 percent of what they learn through the sense of sight? That means if you show something to people, they are far more likely to remember it, at least for a while, than if you tell something to them. Show and tell at the same time and your audience will remember even more.

**When To Use Visuals**

*A picture is worth a thousand words.*

–Chinese Proverb

Nearly any kind of presentation will benefit from some form of visual aid. Shareholders will have a better grasp of earnings or losses when presented with pie charts or bar graphs to show them where the money went. Clients of an advertising agency will have a better understanding of what a new advertising campaign will look and sound like when they are presented with story boards for TV commercials and slides of magazine ads. New hires will catch on to customer relations policies through role-model performances on video. Gardeners will learn how to propagate plants from cuttings when they actually have the plant material in their hands. A prospective customer is more likely to understand your product, and feel the need to buy it, if he or she can see it or touch it. And a message like “sell” or “service” or “quality” takes on greater meaning when it’s projected on a screen or printed on a flip chart.

** **

Sometimes visuals are essential components of a presentation. Examples of times when visuals are a “must” include the following:

Your message is abstract, complex, or difficult to understand.

Your key message or subject is visual in nature.

It is essential that your audience retain your message.

There is controversy or the chance your message could be misinterpreted.

You have more than two or three key points.

You want to add emphasis to a key point.

The presentation includes words or language unfamiliar to the audience.

The presentation is a how-to session involving several steps.

You need to “dress up” a subject that may not be of great interest to the audience.

The presentation includes numbers or mathematical calculations.

**Types of Visuals/Technology**

Visual aids take many forms, for example:

- Flip charts on easels
- Note-book flip charts
- Blackboards with chalk
- Whiteboards with markers
- Overhead transparencies
- Slides
- Videos
- Multimedia productions
- CD ROM
- Computers
- Props
- Three-dimensional models
- Posters
- Banners
- Handouts

And there is enormous diversity for the potential content of your visual aids:

- Photographs
- Typography
- Graphics
- Lists
- Symbols
- Colours
- Shapes
- Charts
- Maps
- Graphs
- Diagrams
- Cartoons

Somewhere among all these possibilities and combinations will be the visual aid that will match your objectives, subject matter, delivery style, audience needs, expectations, and, very importantly, budget.

**2.6 Relations of distance and positions between objects are analysed from different views**.

The difference between the concepts of length and distance is that the length is used to state the length measure of one of the dimensions of a single object, while distance is used to state the length measure that separates two distinct objects.

Understanding size and distance relies upon comparison to a known standard. The comparison might be between objects: a man standing next to a fence in a picture provides a scale for judging the size of the fence. The size of the man is the known standard. Take the man out of the picture and a basis for judgment of the fence size disappears with him.

Distance is measured by comparison to a standard as well. If the man is taller than the house in the picture, we know that the house is far away. Size and distance are shown as relationships by using comparison and contrast. Of course, this method only works if the size of one object is known. A picture of two rocks provides no basis for judgment unless a referent, even a single blade of grass, is added. In the physical world and in pictures of that world, size and distance are measured against known standards. The standard which is most basic for all comparisons is the human body.

**Position and Order**

We also gain meaning from position in a sequence. The order of letters and numbers has meaning: “on” is not “no,” “saw” is not “was,” and “angel” is not “angle.”

The position of numbers tells their value. This relational quality is built into the system. “101 “is different in value from “10.” Learning place value and internalising the meaning of numbers is important. Number position is meaningful in operations like addition, subtraction, multiplication, and division. In fact, each operation has its own set of relational rules or demands. For instance, subtraction requires taking the bottom number from the top number and recognising place values when borrowing.

We describe the position of objects in space relative to other objects with words such as right, left, above, below, and beside.

**2.7 Conjectures As Appropriate To The Situation Are Based On Well-planned Investigations Of Geometrical Properties.**

You cannot speculate regarding geometrical properties, proper measurements are required to determine actual values. Results are achieved through efficient and correct analysis and the manipulation of representations.

**Twenty Conjectures in Geometry:**

*Vertical Angle Conjecture*: Non-adjacent angles formed by two intersecting lines.

*Linear Pair Conjecture*: Adjacent angles formed by two intersecting lines.

*Triangle Sum Conjecture*: Sum of the measures of the three angles in a triangle.

*Quadrilateral Sum Conjecture*: Sum of the four angles in a convex four-sided figure.

*Polygon Sum Conjecture*: Sum of the angles for any convex polygon.

*Exterior Angles Conjecture*: Sum of exterior angles for any convex polygon.

*Isosceles Triangle Conjectures*: Isosceles triangles have equal base angles.

*Isosceles Trapezoid Conjecture*: Isosceles trapezoids have equal base angles.

*Mid-segment Conjectures*: Lengths of mid-segments for triangles and trapezoids.

*Parallel Lines Conjectures*: Corresponding, alternate interior, and alternate exterior angles.

*Parallelogram Conjectures*: Side, angle, and diagonal relationships

*Rhombus Conjectures*: Side, angle, and diagonal relationships.

*Rectangle Conjectures*: Side, angle, and diagonal relationships.

*Congruent Chord Conjectures*: Congruent chords intercept congruent arcs.

*Chord Bisector Conjecture*: The bisector of a chord passes through the centre of the circle.

*Tangents to Circles Conjectures*: A tangent to a circle is perpendicular to the radius*. *

*Inscribed Angle Conjectures*: An inscribed angle has half the measure of intercepted arc.

*Inscribed Quadrilateral Conjecture*: Opposite angles are supplements.

*The Number “Pi” Conjectures*: Circumference and diameter relationship for a circle.

*Arc Length Conjecture*: Formula to calculate the length of an arc on a circle.

**2.8 Representations Of The Problems Are Consistent With & Appropriate To The Problem Context. The Problems Are Represented Comprehensively & In Mathematical Terms.**

**Euler’s Theorem**

You’ve already learned about many polyhedral properties. All of the faces must be polygons. Two faces meet along an edge. Three or more faces meet at a vertex. In this section, you’ll learn about a property of polyhedra known as Euler’s Theorem, because it was discovered by the mathematician Leonhard Euler (pronounced “Oil-er”). You already know that a polyhedron has faces (F), vertices (V), and edges (E). But Euler’s Theorem says that there is a relationship among F, V, and E that is true for every polyhedron. That’s right — every polyhedron, from a triangular prism to a hexagonal pyramid to a truncated hedron.

Euler’s Theorem actually played a role in a notable discovery. In some chemistry experiments, a group of researchers believed that they had found a new molecule with the exact weight of 60 carbon atoms. Although they couldn’t see this molecule, they speculated that its shape was a truncated icosahedron — a “soccer ball” in which 60 carbon atoms (vertices) were joined together by 90 bonds (edges). From Euler’s Theorem, they then knew that the atoms must be arranged to form a spherical soccer ball with 32 faces, some of them hexagons and some pentagons.

**2.9 Results Are Achieved Through Efficient & Correct Analysis & Manipulation Of Representations.**

**Platonic Solids**

In this section on three-dimensional solids, you’ve seen a lot of polyhedra. But there are five special polyhedra — known collectively as the Platonic solids — that are different from all the others. What makes the Platonic solids special? Well, two things, actually.

- They are the only polyhedra whose faces are all exactly the same. Every face is identical to every other face. For instance, a cube is a Platonic solid because all six of its faces are congruent squares.
- The same number of faces meet at each vertex. Every vertex has the same number of adjacent faces as every other vertex. For example, three equilateral triangles meet at each vertex of a tetrahedron.

No other polyhedra satisfy both of these conditions. Consider a pentagonal prism. It satisfies the second condition because three faces meet at each vertex, but it violates the first condition because the faces are not identical — some are pentagons and some are rectangles.

**2.10 Problem-solving Methods Are Presented Clearly, Logically & In Mathematical Terms.**

** ****Mathematical argument and evaluation based on logical deduction.**

**“ Logical deduction is the one and only true powerhouse of mathematical thinking.”**

— Jean Dieudonne

What is logical deduction?

Deduction, in logic, the form of reasoning by which a specific conclusion is inferred from one or more premises. In valid deductive reasoning, the conclusion must be true if all the premises are true. Thus, if it is agreed that all human beings have one head and two arms, and that Bertha is a human being, then it can logically be concluded that Bertha must have one head and two arms. This is an example of a syllogism, an argument in which two premises are given and a logical conclusion is deduced from them. Deduction is often expressed in the form of syllogisms.

It is the most straightforward logical manipulation rule.

Examples

- All men are mortal. Socrates is a man. Therefore Socrates is mortal.
- All feminists are unreasonable. She is a feminist. Therefore she is unreasonable.
- All birds are black. That is a bird. Therefore it is black.

Reflections on the chosen problem solving strategy reveal strengths and weaknesses of the strategy.

Activity 9:Do a SWOT analysis on the problem solving strategy that you have chosen.

**2.12 Alternative Strategies To Obtain The Solution Are Identified & Compared In Terms Of Appropriateness & Effectiveness.**

**Polya’s four steps to problem solving**

If you follow these steps, it will help you become more successful in the world of problem solving.

Polya created his famous four-step process for problem solving, which is used all over to aid people in problem solving:

Step 1: Understand the problem.

Sometimes the problem lies in understanding the problem. If you are unclear as to what needs to be solved, then you are probably going to get the wrong results. In order to show an understanding of the problem, you, of course, need to read the problem carefully. Sounds simple enough, but some people jump the gun and try to start solving the problem before they have read the whole problem. Once the problem is read, you need to list all the components and data that are involved. This is where you will be assigning your variable.

Step 2: Devise a plan (translate).

When you devise a plan (translate), you come up with a way to solve the problem. Setting up an equation, drawing a diagram, and making a chart are all ways that you can go about solving your problem.

Step 3: Carry out the plan (solve).

The next step, carry out the plan (solve), is big. This is where you solve the equation you came up with in your ‘devise a plan’ step.

Step 4: Look back (check and interpret).

You may be familiar with the expression ‘don’t look back’. In problem solving it is good to look back (check and interpret).. Basically, check to see if you used all your information and that the answer makes sense. If your answer does check out, make sure that you write your final answer with the correct labelling.

Example:
In a blueprint of a rectangular room, the length is 1 cm more than 3 times the width. Find the dimensions if the perimeter is to be 26 cm. |

Step 1: Understand the problem. |

Make sure that you read the question carefully several times. We are looking for the length and width of the rectangle. Since length can be written in terms of width, we will let w = width length is 1 cm more than 3 times the width:1 + 3w = length |

**Step 4:** **Look back ****(check and interpret)**.

If width is 3, then length, which is 1 cm more than 3 times the width would have to be 10. The perimeter of a rectangle with width of 3 cm and length of 10 cm does come out to be 26.
FINAL ANSWER: Width is 3 cm. Length is 10 cm |

__Important embedded knowledge__

**Use of the Cartesian co-ordinate system in determining location and describing relationships.**

The modern Cartesian coordinate system in two dimensions (also called a **rectangular coordinate system**) is commonly defined by two axes, at right angles to each other, forming a plane (an xy-plane). The horizontal axis is labelled **x**, and the vertical axis is labelled **y**. In a three dimensional coordinate system, another axis, normally labelled **z**, is added, providing a sense of a third dimension of space measurement. The axes are commonly defined as mutually orthogonal to each other (each at a right angle to the other). (Early systems allowed “oblique” axes, that is, axes that did not meet at right angles.) All the points in a Cartesian coordinate system taken together form a so-called **Cartesian plane**. The point of intersection, where the axes meet, is called the *origin* normally labelled **O**. With the *origin* labelled **O**, we can name the **x** axis **Ox** and the **y** axis **Oy**. The x and y axes define a plane that can be referred to as the **xy** plane. Given each axis, choose a unit length, and mark off each unit along the axis, forming a grid. To specify a particular point on a two dimensional coordinate system, you indicate the **x** unit first (abscissa), followed by the **y** unit (ordinate) in the form **(x,y)**, an ordered pair. In three dimensions, a third **z** unit is added, **(x,y,z)**.The choices of letters come from the original convention, which is to use the latter part of the alphabet to indicate unknown values. The first part of the alphabet was used to designate known values. An example of a point **P** on the system is indicated in the picture below using the coordinate **(5,2)**.

The arrows on the axes indicate that they extend forever in the same direction (i.e. infinitely). The intersection of the two x-y axes creates four quadrants indicated by the roman numerals I, II, III, and IV. Conventionally, the quadrants are labelled counter-clockwise starting from the northeast quadrant. In Quadrant I the values are (x,y), and II:(-x,y), III:(-x,-y) and IV:(x,-y). (see table below.)

Quadrant |
x values |
y values |

I | > 0 | > 0 |

II | < 0 | > 0 |

III | < 0 | < 0 |

IV | > 0 | < 0 |

Three-dimensional coordinate system

Sometime in the early 19th century the third dimension of measurement was added, using the **z** axis.

The coordinates in a three dimensional system are of the form **(x,y,z)**. An example of two points plotted in this system are in the picture above, points **P(5,0,2)** and **Q(-5,-5,10)**. Notice that the axes are depicted in a world-coordinates orientation with the Z axis pointing up.

The x, y, and z coordinates of a point (say P) can also be taken as the distances from the yz-plane, xz-plane, and xy-plane respectively. The figure below shows the distances of point P from the planes.

The three dimensional coordinate system is popular because it provides the physical dimensions of space, of height, width, and length, and this is often referred to as “the three dimensions”. It is important to note that a dimension is simply a measure of something, and that, for each class of features to be measured, another dimension can be added. Attachment to visualising the dimensions precludes understanding the many different dimensions that can be measured (time, mass, colour, cost, etc.). It is the powerful insight of Descartes that allows us to manipulate a multi-dimensional object algebraically, avoiding compass and protractor for analysing in more than three dimensions.

Orientation and “handedness”

The three-dimensional Cartesian coordinate system presents a problem. Once the *x*– and *y*-axes are specified, they determine the line along which the z-axis should lie, but there are two possible directions on this line. The two possible coordinate systems which result are called ‘right-handed’ and ‘left-handed’.

The origin of these names is a trick called the right-hand rule (and the corresponding left-hand rule). If the forefinger of the right hand is pointed forward, the middle finger bent inward at a right angle to it, and the thumb placed a right angle to both, the three fingers indicate the relative directions of the *x*-, *y*-, and *z*-axes respectively in a *right-handed* system. Conversely, if the same is done with the left hand, a left-handed system results.

The right-handed system is universally accepted in the physical sciences, but the left-handed is also still in use.

The left-handed orientation is shown on the left, and the right-handed on the right.

If a point plotted with some coordinates in a right-handed system is replotted with the same coordinates in a left-handed system, the new point is the mirror image of the old point about the xy-plane.

The right-handed cartesian coordinate system indicating the coordinate planes.

More ambiguity occurs when a three-dimensional coordinate system must be drawn on a two-dimensional page. Sometimes the z-axis is drawn diagonally, so that it seems to point out of the page. Sometimes it is drawn vertically, as in the above image (this is called a *world coordinates* orientation).

Further notes:

In analytic geometry the Cartesian coordinate system is the foundation for the algebraic manipulation of geometrical shapes. Many other coordinate systems have been developed since Descartes. One common set of systems use polar coordinates; astronomers often use spherical coordinates, a type of polar coordinate system. In different branches of mathematics coordinate systems can be transformed, translated, rotated, and re-defined altogether to simplify calculation and for specialised ends.

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