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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.continue

Vectors are represented in component form as sum of standard unit vectors multiplied by scalars.



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If a person walks `3` meter north east, `4` meter north. Then, facing north, the person turns clockwise `120^@` and walk `2` meter forward. What is the distance he has to walk to return back to the starting position?vector addition first principle

  • Answer is `3+4+2`
  • answer is not `3+4+2`
  • answer is not `3+4+2`

The answer is 'not `3+4+2`'. The figure shows the problem using line segments and to solve this, trigonometry is used.

vector addition first principle Represent each of the segment in their horizontal and vertical components.
 •  3 meter north east = `3cos45` meter east + `3sin45` meter north

Represent each of the segment in their horizontal and vertical components.vector addition first principle
 •  `3` north east = `3cos45` east + `3sin45` north
 •  `4` north = `4cos90` east + `4sin90` north
 •  `2` at `-30^@` = `2cos(−30)` east + `2sin(−30)` north
 •  Add each of these east and north components individually
 •  `3cos45+4cos90+2cos(−30) = 3.85`
 •  `3sin45+4sin90+2sin(−30) = 5.12`
 •  result distance ` = 3.85i+5.12j`



[Continued..] What made the solution simple?

  • The vector quantities are split into components having same direction
  • The vector quantities are split into components having same direction
  • The vector quantities are split into components having same magnitude

The answer is 'the vector quantities are split into components having same direction' which were easily added.

Note that a vector is given as a sum of components along three dimensions, which is equivalently a sum of scalar multiples of the three unit vectors.

Vector representation is chosen to be sum of standard unit vectors multiplied by scalars.

Vector Component form: Component form of vector representation is the sum of standard unit vector `i`, `j`, `k` multiplied by scalars `a, b, c in RR`.
`vec p = a i + bj + ck`

                            
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