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Thought-Process to Discover Knowledge

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Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

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Introducing Vectors

» Measurement of Quantities

→ Addition: sum of measured amount

→ Subtraction: difference in measured amount

» Some quantities do not add or subtract only in the amount

→ `3` east + `4` north is not `7`

→ `3` east + `4` north-east is approximately `6.48`

» Some quantities cannot be used in calculation if only amount is given without direction

» Measurement of such quantities should capture both the amount and the direction to carryout calculations

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

To completely specify and mathematically use, some quantities require both magnitude (amount) and **direction**.

Measurement of some quantities has both

• amount

• direction

Such quantities are named as **Vectors**.

The quantities those are specified only by 'magnitude' or 'amount' are named as **Scalars**

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To completely specify and to use in mathematical calculations, some quantities require both magnitude and direction.

Starting on learning "difference between vectors and scalars". ;; To completely specify and to use in mathematical calculations, some quantities require both magnitude and direction.

One of the fundamental aspects of science is to measure and specify quantities. Examples:

• mass of an object: `20` gram

• temperature of water: `30^@` Celsius

• the amount of electric current: `5` ampere

• the amount of time taken: `3` seconds

• the amount of distance traveled: `20` meter

Which one in the following is a measurement?

- predecessor of `7` is `6`
- length of a pen is `10` cm
- `3` is subtracted from `4`
- `3` is added to `4`

The answer is ' Length of a pen is `10` centimeters'.

Examples for addition of quantities

• `20` gram mass and `12` gram mass together weigh `32` gram

• `30^@` Celsius heated up to add another `7^@` Celsius make `37^@` Celsius

• `5` ampere current with another `1.2` ampere current added makes `6.2` ampere

• `3` second walk and another `5` second walk make total `8` second walk

• `20` meter distance walked and another `48` meter distance make `68` meter walked in total

Subtraction of quantities

• `3`gm mass is taken away from `20`gram -- results in `17` gram

• `30^@` Celsius cooled down by `38^@` -- results in `-8^@` Celsius

• `5` ampere current source has a `20` ampere current flowing in opposite direction will make `-15` ampere

• planning a `3` second walk and then reducing the time by `1.2` second result in `1.8` second walk

• planned `20` meter distance walk but stopped `5` meters before the end point, this results in `15` meter distance

With this basic of

• measuring quantities,

• addition and subtraction of them

We have a question.

• Is amount of a quantity sufficient to completely specify quantities?

To understand the question, Let us look at another problem.

A person walks in a football ground. At first he walked `3` meters and then another `4` meters. How far he is from the starting point?

- `3 + 4 = 7`
- `4 − 3 = 1`
- `sqrt(3^2+4^2)`
- can-not calculate

The answer is 'cannot calculate' as the information is incomplete.

It is not possible to find the distance in that question. Can you guess why?

- because the direction is not given
- because `3` meters is very long distance to walk

Because whether the person continues in the same direction or the person changes direction in between is not given.

If the person walks `3` meters straight and continues in the same direction for another `4` meters, then How far he is from the starting point?

- `3 + 4 = 7`
- `4 − 3 = 1`
- `sqrt(3^2+4^2)`
- can-not calculate

The answer is '`3 + 4 = 7`', as the distances are added when the directions are same.

If the person walks `4` meters straight. He turns around towards the starting point and walks `3` meters. How far he is from the starting point?

- `3 + 4 = 7`
- `4 − 3 = 1`
- `sqrt(3^2+4^2)`
- can-not calculate

The answer is `4 − 3 = 1`', as the person walks back `3` meter in the opposite direction.

Let us take a person walking from position 'p', `3` meter towards east and another `4` meter towards north. What is the straight distance between the end position and the start position?

- `3 + 4 = 7`
- answer is NOT `3 + 4`

The answer is 'NOT `3 + 4`' . Let us see how to solve.

Let us take a person walking from position 'p', `3` meter towards east and another `4` meter towards north. What is the straight distance between the end position and the start position? Considering this to be a line of `3` meter and another line of `4` meter perpendicular to the first line, we can solve this using ...

- Pythagoras Theorem
- Formula to find Hypotenuse of right angled triangle
- both the above

Answer is 'both the above', as Pythagoras Theorem helps in finding hypotenuse of a right angle triangle using the two sides.

`3` meter and `4` meter are two sides of a right angled triangle and so

`text(distance) = sqrt(3^2+4^2)

Apart from the 'amount' of distances the person is walking, it is important to note down the 'direction' in which the person is walking.

Measurement of some quantities require not just the **amount **, also the **direction**.

Does 'vector' have a meaning?

- Yes. It means 'to direct towards a point'.
- No. It is purely a mathematical term.

Answer is, Vector means 'to direct towards a point'.

Like in the sentence, 'the aeroplane is vectored towards New York.'

are there any quantities that are measured with only 'magnitude' and do not have 'direction'?

- Yes
- No

Answer is 'Yes', There are quantities those do not have direction information when measured.

Identify the quantity that does not have 'direction' information.

- Mass of an object
- temperature of water
- the amount of time taken
- All the above

All the three given are examples of quantities without direction information.

What does the word 'scalar' mean?

- It is derived from 'to scale something'; meaning to measure.
- It does not have any meaning.

Answer is 'Scalar is derived from to scale ; to measure'. It is used in mathematics to denote quantities with only magnitude.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**magnitude:** The measure of amount of a quantity.

**Direction: ** The measure of relative position or orientation of a quantity.

**Vector: ** Quantity that requires both magnitude and direction information to process mathematically.

**Scalar: ** Quantity that requires only magnitude information to process mathematically.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Identify the statement in which direction information is specified.

- he walks on grass
- he walks along with his friend
- he walks towards east
- he walks towards his success

Answer is 'he walks towards east' - the statement specifies direction 'east'.

Identify the statement in which direction information is specified.

- She felt the gravity pulling down
- She understood the gravity of the situation
- Gravitational force in Earth is higher than that of Moon

Answer is 'She felt the gravity pulling down' – the statement specifies direction 'downwards'.

Identify the statement in which direction information is specified.

- friction makes it possible to walk
- friction acts against the movement
- friction causes wear

Answer is 'friction acts against the movement' - the statement specifies direction 'against the movement'

Identify the statement in which direction is specified.

- On hearing the news, he smiled.
- He spilled coffee.
- He throws the ball towards me.

Answer is 'He throws the ball towards me' – the statement specifies direction 'towards' a reference.

Identify the statement in which direction is specified.

- wind blew at `45^@`angle
- wind uprooted a tree weighing 250kg
- wind caused the fan to rotate fast

Answers is 'wind blew at `45^@`angle' - the statement specifies direction 'at `45^@`angle'.

Identify the statement in which direction is specified.

- the boat sails along with the flow of the river
- the boat carries 200 kg wheat
- the boat weighs 40 kg

Answer is 'the boat sails along with the flow of the river' - the statement specifies direction 'along with the flow'.

What is a scalar quantity?

- A quantity without direction
- A quantity with direction

Answer is 'A quantity without direction', A scalar quantity does not have direction as part of the measurement.

What is a vector quantity?

- A quantity without direction
- A quantity with direction

Answer is 'A quantity with direction'. When measured, a vector quantity is specified by both magnitude and direction.

A person, facing east, measures the temperature of the glass of water as `25^@`*C*. Is this measure scalar or vector?

- Facing East and `25^@`
*C*- so a vector - Measure is just `25^@`
*C*- so a scalar

Correct answer - Measure is `25^@`*C* - so a scalar. The person facing East does not add to the measure, rather an irrelevant information.

A person, measures the distances between three points on the floor as *AB* `= 20`*cm*, *BC* `= 10`*cm*, and *AC* `= 15`*cm*. Are these measurements scalars or vectors?

- No direction information given - so scalars
- the points on floor forms a triangle - so vectors

Answer - The points on the floor forms a triangle and direction information can be derived as angle between line segments. So, these are implicitly vectors with only magnitude of them are provided.

A person, measures the distances between two points on the floor as *AB* `= 20`*cm* and *BC* `= 10`*cm*. Are these distances scalars or vectors?

- These are supposed to be vector quantities, but only the magnitudes are measured.
- These are not vector quantities as the direction information is not available.

Answer is - These are supposed to be vector quantities. The direction information is not measured. This measurement is incomplete.

*your progress details*

Progress

*About you*

Progress

One of the fundamental aspects of science is to measure and specify quantities. Examples ;; mass of an object: 20 gram ;; temperature of water: 30 degree Celsius ;; the amount of electric current: 5 ampere ;; the amount of time taken: 3 seconds ;; the amount of distance traveled: 20 meter

Which one in the following is a measurement?

predecessor;7;6

predecessor of 7 is 6

length;pen;centi;meter

length of a pen is 10 centi-meter

subtracted;from

3 is subtracted from 4

3;added;4

3 is added to 4

The answer is ' Length of a pen is 10 centimeters'.

Examples for addition of quantities;; 20 gram mass and 12 gram mass together weigh 32 gram;; 30 degree Celsius heated up to add another 7 degree Celsius make 37 degree Celsius ;; 5 ampere current with another 1.2 ampere current added makes 6.2 ampere ;; 3 second walk and another 5 second walk make total 8 second walk ;; 20 meter distance walked and another 48 meter distance make 68 meter walked in total

Subtraction of quantities; 3 gram mass is taken away from 20 gram -- results in 17 gram ;; 30 degree Celsius cooled down by 38 degree results in -8^@ Celsius ;; 5 ampere current source has a 20 ampere current flowing in opposite direction will make -15 ampere ;; planning a 3 second walk and then reducing the time by 1.2 second result in 1.8 second walk ;; planned 20 meter distance walk but stopped 5 meters before the end point, this results in 15 meter distance.

With this basic of ;; measuring quantities, ;; addition and subtraction of them ;; We have a question.; Is amount of a quantity sufficient to completely specify quantities? To understand the question, Let us look at another problem.

A person walks in a football ground. At first he walked 3 meters and then another 4 meters. How far he is from the starting point?

plus;+

3 + 4 = 7

minus;-

4 − 3 = 1

square;squared;root

square root of 3 squared plus 4 squared

can;not;calculate

can-not calculate

The answer is 'cannot calculate' as the information is incomplete.

It is not possible to find the distance in that question. Can you guess why?

direction;not;given

because the direction is not given

3;meter;very;long

because 3 meters is very long distance to walk

Because whether the person continues in the same direction or the person changes direction in between is not given.

If the person walks 3 meters straight and continues in the same direction for another 4 meters, then How far he is from the starting point?

plus;+

3 + 4 = 7

minus;-

4 − 3 = 1

square;squared;root

square root of 3 squared plus 4 squared

can;not;calculate

can-not calculate

The answer is ' 3 + 4 = 7 ', as the distances are added when the directions are same.

If the person walks 4 meters straight. He turns around towards the starting point and walks 3 meters. How far he is from the starting point?

plus;+

3 + 4 = 7

minus;-

4 − 3 = 1

square;squared;root

square root of 3 squared plus 4 squared

can;not;calculate

can-not calculate

The answer is 4 − 3 = 1 ', as the person walks back 3 meter in the opposite direction.

Let us take a person walking from position 'p', 3 meter towards east and another 4 meter towards north. What is the straight distance between the end position and the start position?

7;equals;equal

3 + 4 = 7

not;answer

answer is NOT 3 + 4

The answer is 'NOT 3 + 4 ' . Let us see how to solve.

Let us take a person walking from position 'p', 3 meter towards east and another 4 meter towards north. What is the straight distance between the end position and the start position? ;; Considering this to be a line of 3 meter and another line of 4 meter perpendicular to the first line, we can solve this using ;;

Pythagoras

Pythagoras Theorem

Formula;find;hypotenuse;right;triangle

Formula to find Hypotenuse of right angled triangle

both;above

both the above

The answer is "both the above" as Pythagoras Theorem helps in finding hypotenuse of a right angle triangle using the two sides. ;; 3 meter and 4 meter are two sides of a right angled triangle and so distance = square root of 3 squared plus 4 squared.

Apart from the 'amount' of distances the person is walking, it is important to note down the 'direction' in which the person is walking. ;; Measurement of some quantities require not just the amount , also the direction.

To completely specify and mathematically use, some quantities require both magnitude or amount and direction.

Magnitude: The measure of amount of a quantity.

Direction : The measure of relative position or orientation of a quantity.

Identify the statement in which direction information is specified.

on;grass

he walks on grass

along;with;friend

he walks along with his friend

towards;east

he walks towards east

towards;success

he walks towards his success

Answer is 'he walks towards east' - the statement specifies direction 'east'.

Identify the statement in which direction information is specified.

pulling;down;felt

She felt the gravity pulling down

understood;situation

She understood the gravity of the situation

higher;earth;force;moon;than

Gravitational force in Earth is higher than that of Moon

Answer is 'She felt the gravity pulling down' – the statement specifies direction 'downwards'.

Identify the statement in which direction information is specified.

makes;possible;walk

friction makes it possible to walk

acts;against;movement

friction acts against the movement

causes;wear

friction causes wear

Answer is 'friction acts against the movement' - the statement specifies direction 'against the movement'

Identify the statement in which direction is specified.

hearing;news;smiled

On hearing the news, he smiled.

coffee;spilled

He spilled coffee.

throws;ball;towards;me

He throws the ball towards me.

Answer is 'He throws the ball towards me' – the statement specifies direction 'towards' a reference.

Identify the statement in which direction is specified.

blew;at;45;degree;angle

wind blew at 45 degree angle

uprooted;tree;weighing;250

wind uprooted a tree weighing 250 kilograms

caused;fan;rotate;fast

wind caused the fan to rotate fast

The answer is "wind blew at 45 degree angle" - the statement specified direction ; at 45 degree angle.

Identify the statement in which direction is specified.

sails;along;with;flow;water

the boat sails along with the flow of the river

carries;200;wheat

the boat carries 200 kg wheat

weighs;40

the boat weighs 40 kg

Answer is 'the boat sails along with the flow of the river' - the statement specifies direction 'along with the flow'.

Measurement of some quantities has both ; amount ;; direction ;; Such quantities are named as Vectors.

Vector: Quantity that requires both magnitude and direction information to process mathematically.

Does 'vector' have a meaning?

yes;s;means;direct;towards;point

Yes. It means 'to direct towards a point'.

no;purely;mathematical;term

No. It is purely a mathematical term.

Answer is, Vector means 'to direct towards a point'.

Like in the sentence, 'the aeroplane is vectored towards New York.'

Like in the sentence, 'the aeroplane is vectored towards New York.'

are there any quantities that are measured with only 'magnitude' and do not have 'direction'?

yes;s

Yes

no

No

Answer is 'Yes', There are quantities those do not have direction information when measured.

Identify the quantity that does not have 'direction' information.

mass;object

Mass of an object

temperature;water

temperature of water

amount;time;taken

the amount of time taken

all;above

All the above

All the three given are examples of quantities without direction information.

The quantities those are specified only by 'magnitude' or 'amount' are named as Scalars.

Scalar: Quantity that requires only magnitude information to process mathematically.

What does the word 'scalar' mean?

derived;scale;something;measure

It is derived from 'to scale something'; meaning to measure.

not;have;any

It does not have any meaning.

Answer is 'Scalar is derived from to scale ; to measure'. It is used in mathematics to denote quantities with only magnitude.

What is a scalar quantity?

without

A quantity without direction

with

A quantity with direction

Answer is 'A quantity without direction', A scalar quantity does not have direction as part of the measurement.

What is a vector quantity?

without

A quantity without direction

with

A quantity with direction

Answer is 'A quantity with direction'. When measured, a vector quantity is specified by both magnitude and direction.

A person, facing east, measures the temperature of the glass of water as 25 degree Celsius. Is this measure scalar or vector?

facing;east;vector

facing east and 25 degree Celsius - so a vector

measure;just;scalar

measure is just 25 degree Celsius - so a scalar

The answer is "measure is just 25 degree Celsius - so a scalar". The person facing East does not add to the measure, rather an irrelevant information.

A person, measures the distances between three points on the floor as A B = 20 centimeter, B C = 10 centimeter, and A C = 15 centimeter. Are these measurements scalars or vectors?

no;direction;given;scalars

No direction information given - so scalars

points;on;floor;triangle;vectors

the points on floor forms a triangle - so vectors

Answer - The points on the floor forms a triangle and direction information can be derived as angle between line segments. So, these are implicitly vectors with only magnitude of them are provided.

A person, measures the distances between two points on the floor as A B = 20 centimeter, and B C = 10 centimeter. Are these distances scalars or vectors?

supposed;but;only;magnitudes

These are supposed to be vector quantities, but only the magnitudes are measured.

not;direction;available

These are not vector quantities as the direction information is not available.

Answer is - These are supposed to be vector quantities. The direction information is not measured. This measurement is incomplete.