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Thought-Process to Discover Knowledge

Welcome to nubtrek.

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.
mathsVector AlgebraIntroduction to Vector Quantities

### Introducing Vectors

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

click on the content to continue..

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
• 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

•  3gm mass is taken away from 20gram -- 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
• 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 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
• 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
• 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
• 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
• 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.

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

magnitude: The measure of amount of a quantity.

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

Solved Exercise Problem:

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 east
• he walks towards his success

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

Solved Exercise Problem:

Identify the statement in which direction information is specified.

• She felt the gravity pulling down
• 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'.

Solved Exercise Problem:

Identify the statement in which direction information is specified.

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

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

Solved Exercise Problem:

Identify the statement in which direction is specified.

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

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

Solved Exercise Problem:

Identify the statement in which direction is specified.

• wind blew at 45^@angle
• 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'.

Solved Exercise Problem:

Identify the statement in which direction is specified.

• the boat sails along with the flow of the river
• 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'.

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

In "number systems", we have learned the following.

•  Whole numbers

•  Integers

•  Fractions and Decimals

Let us revise these in few questions.

What are whole numbers?

• whole numbers are used to count objects
• whole numbers are 0,1,2,cdots
• both the above
• both the above

The answer is "both the above".

What are integers?

• integers are directed whole numbers
• integers are directed whole numbers
• integers are part of a whole

The answer is "integers are directed whole numbers".

Whole numbers representation is not sufficient to represent directed numbers.

For example, consider the numbers in
• I received 3 candies and
& bull; I gave 3 candies.

In the whole numbers, both these are represented as 3.

In integers, the first is +3 and the second is -3.

Integer numbers are represented as follows.

3 is represented as either text(received:)3 or text(aligned:)3.

-3 is represented as either text(given:)3 or text(opposed:)3.

What are fractions?

• fractions are directed whole numbers
• fractions are numbers representing part of a whole
• fractions are numbers representing part of a whole

The answer is "fractions are numbers representing part of a whole".

What are decimals?

• decimals are fractions in a standard form
• decimals are fractions in a standard form
• decimals and fractions are not related

The answer is "decimals are fractions in a standard form".

Whole numbers and Integers representation is not sufficient to represent quantities of part of an object.

For example, A pizza is cut into 8 pieces.

3 whole pizzas and 5 pieces of a cut pizza are remaining.

Whole numbers or integers represent them as two quantities: 3 pizzas and 5 pieces when one whole is cut into 8 pieces. This representation is descriptive.

The same in fractions is 3 5/8.

The same in decimals is 3.625

It is noted that the direction specified in the number system refers to

•  positive OR text(aligned:)

•  negative OR text(opposed:)

The Integers are directed-whole-numbers, having positive integers and negative integers.

The fractions are also in directed form, having positive fractions and negative fractions. Likewise for decimals, positive decimals and negative decimals.

The direction information of vectors refers to the spatial direction,

•  x-axis (eg: right and left for a person)

•  y-axis (eg: forward and backward for a person) and

•  z-axis (eg: up and down for a person)

Note, the right-left, forward-backward, up-down are one particular example. It can also be east-west, north-south, up-down or other forms.

The values of a vector along the given spatial directions can be positive or negative (meaning aligned or opposed in direction).

Hope you understand the difference between

•  direction specified for positive and negative numbers. And

•  the spatial direction specified for vectors in 2-dimensions or 3-dimensions

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

Does the word 'vector' have a meaning?

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

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

Solved Exercise Problem:

What is a scalar quantity?

• A quantity without direction
• 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.

Solved Exercise Problem:

What is a vector quantity?

• A quantity without direction
• A quantity with direction
• A quantity with direction

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

Solved Exercise Problem:

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
• 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.

Solved Exercise Problem:

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

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

Solved Exercise Problem:

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

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

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