Vectors and Scalars
What is a vector, how is it different from a scalar?.........................................................
Which mathematical operations can you perform with vectors?............................
Listening https://ocw.mit.edu/courscs/matheniatics/18-02-muItivariable-caicuius-faU-2007/video-lectures/lecture-l-dot-product/
Listen for the first time to the lecture and decide whether the statements are true or false.
What is the nationality of the speaker?
1) Almost all students have heard about vectors before.
2) In the first week there will not be many new things to learn.
3) If the students have problems with vectors, they can go to the instructor's house and ask him.
4) Vector has both the direction and size.
5) If we are in the plane, we use x-y-z axis.
6) Vector quantity is indicated by an arrow above.
7) In the textbooks vectors are in bold because it is easier to read.
8) A vector <j> hat points along the z axis and has length one.
9) The notation al and a2 is in angular brackets.
10) The length of a vector is a scalar quantity.
Listen for the second time to the lecture and try to continue the sentences.
a) So let's start right away with stuff that we will need to see before we can go on to
b) And, so, now, the way that we compute things with vectors, typically, as we introduce
c) When we have a vector quantity,
d) So, the length of a vector we denote by, if you want,
e) OK, so,.....................................................as opposed to a vector quantity.
f) But, a vector doesn't really have, necessarily,
Unit9 Measurements
Ratio and Proportion
Example:
(6.5,4,3,2,1) is an arithmetical progression because each dement is formed by subtracting 1 from the preceding
element.
a) 1,2.3,4,5,6
b) 2,4,8,16,32
12. Solve these problems:
d) x,(x4-a),(x+2a),(x + 3a)
e) tit 16.21,26.3a f.     x x x x
a a'' a3 a*
Section 1 Presentation
l. Look at diese examples:
a} An arithmetical progression begins i ^,......What are the next
two terms?
b) A geometrical progression begins\, $.......What are the next two
terms?
Usit 4~- r\ "*\
Vectors and scalers      (/C /'
A car is travelling north aldbgi road at 60 km h "1. We say that it has a velocity of 60 km h ~1 north. It has a speed of 60 km h "1. The second quantity, speed, is a scalar quantity, that is, it is a size, or magnitude. The first quantity, velocity, is a vector, that is, it has both magnitude and direction. ,
The magnitude of a quantity is usually expressed in relation to a standard unit of magnitude. For example, if the mass of a metal cylinder is 2 500 g, then it is 2500 times the unit of mass, that is, the gramme. If the electrical power of a light bulb is 40 W, then it is 40 times the unit of power the watt. Both of these are scalar quantities.
If a certain town is 150 km from London, then the distance from London to the town is 150000 times the unit of distance, the metre. Distance, again, is a scalar quantity. Now, if we want to know the exact location of the town, we also need to know the direction, that is, we need the vector quantity displacement, which consists of both distance and direction, I SO km north west,
14. Divide the following quantities into vector or scalw quantities:
speed, mass, displacement, weight, force, acceleration, velocity, distance, volume, temperature, momentum, power
15. Say whether the following statements are true or false. Correct the false statements.
a) The mass of an object is the same as the weight of the object.
b) An ordinary number is a vector quantity.
c) A vector quarftky consists of two parts.
d) A position in the Cartesian co-ordinate system may be expressed by a vector.
b     AB is approximately three times as long as CD.
C   The area of rectangle ABCD is exactly twice as big as that of A ABE.
Now make similar sentences about the following:
16. PVZZU-
Comment on this statement:
A train travels 60 km at 30 km b"
average speed is therefore 45 lem h~
then 60 km at 60 km h~l. Its
a) ABC......ACB.
b) PR......XZ.
c) APQR......AXYZ.
d) ABCD......the inscribed
semicircle.
c) ......any circle......the inscribed
square,
f) The circumference......the
diameter.
g) ......rhombus ABCD.
ABE.
h) ......cylinder......cone with the
same base and height
i) ......cube A......cubeB.
The lengths of AB and CD are in the ratio of approximately 3:1.
The areas of ABCD and ABE are in the ratio of exactly 2:1.
Mow make similar sentences about the examples in exercise 1.
Section'2' Development
3. Look and read:
This bar graph shows the relative sizes of some countries in millions of square kilometres.
• The USSR is far larger than India,
• India is considerably larger than France.
• Canada is slightly larger than China.
USSR     Cbirta Franca Canada India
Now compare the other countries in the same way.
4. Look at the table below showing the heights of the highest mountains hi the different continents. Draw a bar graph to Slostrate the heights and then compare the heights of different mountains:
Continent Africa Antarctica Asia . . Australasia Europe
North America South America
5. Look and read: Variation
Highest Mountain
Kilimanjaro
Vinson Massif
Everest
Wilhelm
Elbrus
McKinley
Aconcagua
Height (in metres)
5963
5139
8880
4693
5633
6187
6959
yccx     The ratio between y and x is a constant y is directty proportional to x. We say that y varies directly as x.
y<=c-
y QCX2
y is directly proportional to the reciprocal of x.
We say that y is inversely proportional to x and that y varies
inversely as x.
y is directly proportional to the product of x and z.
We say that y is jointly proportional to x and z, and that y
variesJointly as x and z.
Describe the relationship between the following quantities, where k is a constant:
a) y = £ ' J x
b) y = kxz
c) y:x = k
d) y=*kx
e) y:Uk 6. Look and read:
			arc;
			
! ;			
The volume of a gas is inversely proportional to its pressure. The smaller the volume, the higher the pressure.
Now make similar sentences about the following:
a) the density and pressure of a gas (for a constant temperature)
b) the volume and temperature of a gas (for a constant pressure)
c) the velocity of a falling body and the time it has been falling
d) acceleration and mass for a constant force
e) the electrical resistance of a wire and its cross-sectional area f} the circumference of a circle and its diameter
g) the volume of a cylinder and its cross-sectional area and height
Section 3 Reading
7. Read this:
The ratio of two quantities is the magnitude of one quantity relative to
the other. Division of the quantity a by the quantity b gives the ratio ?.
whkh can also be written as a: b and is read as 'the ratio of a to b". For example, the ratio of boys to girls in a particular school is 3:2. If the
^vuyw^^JvjiMjpils, then we can see that | of these are boys and | < gi^^c, iere arc 150 boys and 100 girls.
Relative sizes of more than two quantities may be expressed by ratio. For example, the ratio of AB:BC*.AC in triangle ABC is 3:4:6. Hence we can see that AC is twice as long as AB. Any triangle which is similar to triangle ABC has sides in exactly the same ratio.
When the ratio of one pair of quantities is equal to the ratio of another pair of quantities, the two pairs are said to be in proportion. If we say
that a,b,c,d, are in proportion, we mean that g ~ 3- A- property of this
proportion is that the reciprocals are also in proportion. Moreover, the ratio of the numerators is equal to the ratio of the denominators.
Say whether the following statements are true or false. Correct the false statements.
a)
b)
c)
d)
e)
In a test a student scored 30 out of 100. This gives a ratio of correct answers to incorrect answers of 3:10.
The length of the shortest side of a triangle similar to triangl e ABC is 12 cm. The other sides are therefore 16 and 24 cm long. Any triangle with sides in the ratio 2 ;3:6 is a right-angled triangle. We may use ratios to express the relationship between any number of quantities.
If a, b, c, d are in proportion, then ad = be.
8. Complete these exercises:
The last two sentences of the reading passage contain two properties of the proportion 5 = g- Put these two properties into mathematical form, and number them 1 and 2.
In the calculations which follow, there are four more properties of the proportion. Complete the proof by adding expressions from the lists.
From properties 2 and 3, we" have Adding 1 to each side, we obtain From properties 2 and 5, we have Similarly
From properties 2 and 4, we have
But
Therefore "Given
Substituting, we obtain
art a-c/
b)
■5+1-1+1
c) ......^«landj^l
d) ......*±5- ^Property 3)
e) ......1=£ = £^ (Property 4)
. a-b b 8) ......č=d" = 3
h) ......|±§ =      (Property 5)
i) ......J±| = |±^(Property6)
Section 4 Listening
Using percentages hi statistics
9. Lbten to the passage and write down these words in the order in which you hear them:
passed respectively while
whereas relative higher
improvement       yesterday meat
10. Copy the following diagrams, which iuostrate the statistics in the fire examples:
The five examples given in the passage may be. summarised as: (1) bread; (2) wages; (3) exam;<4) restaurant; (5) meat.
vivau »      ** w * —\ • f ----------- ■ -> *
a) Put the correct title under each diagram.
b) Label the diagrams as accurately as possible a$i
using the figures