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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
mathsVector AlgebraIntroduction to Vector Quantities

Mathematical Representation of Vectors

Vectors are quantities with magnitude and direction.

How are the vectors represented mathematically?

Understanding that is the objective of this session.



click on the content to continue..

Identify one quantity with direction in the following.

  • `3` meter long
  • `3` meter North
  • `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?

  • `30`km away
  • `30`km away at `4`km altitude above sea level
  • `30`km to the east, `40`km north, and at `4`km altitude above sea level
  • `30`km to the east, `40`km north, and at `4`km 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
`30`km East + `40`km North + `4`km Altitude

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

Do you understand the figure?mathematical representation of vectors 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)`
  • `(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.mathematical representation of vectors 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.

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 `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.

Solved Exercise Problem:

Represent OP as a vector.practice representing a 2d vector

  • `3i+5j`
  • `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`.

Solved Exercise Problem:

What is the vector form of `bar(OP)`?practice representing a 3d vector

  • `6i+3j+4k`
  • `3i+6j+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

Solved Exercise Problem:

What is the vector form of OP?practice representing a 3d vector

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

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

Solved Exercise Problem:

What is the vector form of OP?practice representing a 2d vector

  • `2.4i+ 7/2 j`
  • `2.4i+ 7/2 j+0k`
  • both the above
  • 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.

Solved Exercise Problem:

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`
  • `2i-1.2j+1.4k`

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

Solved Exercise Problem:

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
  • all the above

answer is 'all the above'

                            
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