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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.

User Guide

Welcome to nubtrek.

The content is presented in small-focused learning units to enable you to
think,
figure-out, &
learn.

Just keep tapping (or clicking) on the content to continue in the trail and learn.

User Guide

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

User Guide

nub is the simple explanation of the concept.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about.

User Guide

trek is the step by step exploration of the concept.

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step.

User Guide

jogger provides the complete mathematical definition of the concepts.

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen.

User Guide

exercise provides practice problems to become fluent in the concepts.

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge.

summary of this topic

### Introduction to Vector Quantities

Voice

Voice

Home

Representation of Vectors

»  Specify amount or magnitude and direction

→  3m east and 4m north

→  3m east and 4m north-east

»  How were the quantities added?

→  3m east + 4m north = sqrt(3^2+4^2) as per Pythagoras theorem

→  3m east + 4m north-east : Making this easy to do is the objective of "vector algebra"

»  Position of an airplane ai+bj+ck

→  3Okm east (x-axis) = 30i

→  40km north (y-axis) = 40j

→  4km altitude (z-axis) = 4k

»  component form of a vector ai+bj+ck
→  i,j,k represent the direction along x, y, and z axes

→  a, b, c represent the amount or magnitude along the given directions

### Mathematical Representation of Vectors

plain and simple summary

nub

plain and simple summary

nub

dummy

A vector quantity is represented with 3 components along the three directions of 3D coordinates.

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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Vectors are quantities with magnitude and direction.

How are the vectors represented mathematically?

Understanding that is the objective of this session.

Keep tapping on the content to continue learning.
Starting on learning "representation of vectors". ;; vectors are quantities with magnitude and direction. How are the vectors represented mathematically? ;; Understanding that is the objective of this session.

Identify one quantity with direction in the following.

• 3 meter long
• 3 meter North
• 3 meter rope
• 3 meter width

Answer is '3 meter North' - In this the direction 'North' is given.

If someone is calculating the position of an airplane, what would be a reasonable specification of the position?

• 30km away
• 30km away at 4km altitude above sea level
• 30km to the east, 40km north, and at 4km altitude above sea level

Answer is to specify - North, East, and Altitude. With these three pieces of information, the position of the airplane is specified clearly.

We have arrived at representing vectors that have magnitude and direction.

The representation is
30km East + 40km North + 4km Altitude

This can be generalized to a 3D space with three axes x-axis, y-axis, and z-axis. The figure represents x-axis, y-axis, and z-axis. A point 'p' is shown.

Do you understand the figure? What is the x, y, and z coordinates, given in (x,y,z) order, of the point given in figure?

• (7,4,10)
• (4,7,10)
• (10,7,4)
• (4,10,7)

The answer is '(4,7,10)'.
The point is projected onto x-y plane. That is further projected onto x and y axes.

The airplane can be considered as the point 'p' in the three dimensional space. The representation is simplified as
4i+7j+10k
or alternatively
4 hat i+7 hat j+10 hat k
Where
i or hat i represents the direction in x-axis
j or hat j represents the direction in y-axis
k or hat k represents the direction in z-axis

Note: hat i is pronounced as i-hat or i-cap.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

Mathematical representation: A vector quantity is represented in the form ai+bj+ck
or alternatively
a hat i+b hat j+c hat k
Where i, j, k are the directions along x, y, and z axes
and a, b, c are the magnitude along the directions respectively.

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

Represent OP as a vector.

• 3i+5j
• 8
• 5i+3j
• sqrt(3^2+5^2)

answer is '3i+5j'.
The x-axis component is represented with an i and y-axis component is represented with a j.

What is the vector form of bar(OP)?

• 6i+3j+4k
• 3i+6j+4k
• 4i+3j+6k
• 4i+6j+3k

answer is '3i+6j+4k'. Referring to the figure, the component along each of the axes
3i: 3 along x axis
6j: 6 along y axis
4k: 4 along z axis

What is the vector form of OP?

• -11/2 i-1.3j+4k
• 4i-11/2 j-1.3k
• -1.3i-11/2 j-4k
• -1.3i-11/2 j+4k

answer is '-1.3i-11/2 j+4k'.

What is the vector form of OP?

• 2.4i+ 7/2 j
• 2.4i+ 7/2 j+0k
• both the above
• none of the above

answer is 'both the above'. When a two dimensional vector is presented, the component along the third dimension is 0.

A point is located from the x, y, z-axes at distances 2, -1.2, and 1.4 units respectively. What is the vector representation of the point?

• 2−1.2+1.4
• 2.2
• sqrt(2^2+1.2^2+1.4^2)
• 2i-1.2j+1.4k

answer is '2i-1.2j+1.4k'

What does a vec(OP) = −.5i+2.1j−.6k mean?

• the point P is −0.5 unit away from x-axis
• the point P is 2.1 unit away from y-axis
• the point P is −0.6 unit away from z-axis
• all the above

Progress

Progress

Identify one quantity with direction in the following.
long
3 meter long
north
3 meter North
rope
3 meter rope
width
3 meter width
Answer is ' 3 meter North' - In this the direction 'North' is given.
If someone is calculating the position of an airplane, what would be a reasonable specification of the position?
1
2
3
Answer is to specify - North, East, and Altitude. With these three pieces of information, the position of the airplane is specified clearly.
We have arrived at representing vectors that have magnitude and direction. ;; The representation is : 30 kilometer East + 40 kilometer North + 4 kilometer Altitude
This can be generalized to a 3D space with three axes x-axis, y-axis, and z-axis.;; The figure represents x-axis, y-axis, and z-axis. A point p is shown.
Do you understand the figure? What is the x, y, and z coordinates, given in (x,y,z) order, of the point given in figure?
1
2
3
4
The answer is "4, 7, 10". The point is projected onto x-y plane. That is further projected onto x and y axes.
The airplane can be considered as the point p in the three dimensional space. ; The representation is simplified as ;; 4i+7j+10k ;; or alternatively ;; 4 i cap +7 j cap+ 10 k cap;; Where i or i cap represents the direction in x-axis ;; j or j cap represents the direction in y-axis ;; k or k cap represents the direction in z-axis
A vector quantity is represented with 3 components along the three directions of 3D coordinates.
Mathematical representation: A vector quantity is represented in the form a i+b j+c k ;; or alternatively ;; a i cap+b j cap+c k cap ;; Where i, j, k are the directions along x, y, and z axes ;; and a, b, c are the magnitude along the directions respectively.
Represent OP as a vector.
1
2
3
4
The answer is "3 i + 5 j ". The x-axis component is represented with an i and y-axis component is represented with a j.
What is the vector form of O P?
1
2
3
4
The answer is "3 i + 6 j + 4 k". Referring to the figure, the component along each of the axes. 3 i represents 2 along x axis. 6 j represents 6 along y axis. 4 k represents 4 along z axis.
What is the vector form of O P?
1
2
3
4
The answer is "minus 1 point 3 i minus 11 by 2 j plus 4 k"
What is the vector form of O P?
1
2
3
4
answer is 'both the above'. When a two dimensional vector is presented, the component along the third dimension is 0.
A point is located from the x, y, z-axes at distances 2, minus 1 point 2, and 1 point 4 units respectively. What is the vector representation of the point?
1
2
3
4
The answer is "2 i minus 1 point 2 j + 1 point 4 k"
what does a vector O P = minus point 5 i + 2 point 1 j minus point 6 k mean?
1
2
3
4