nubtrek

Server Error

Server Not Reachable.

This may be due to your internet connection or the nubtrek server is offline.

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

In this page, the basic mathematical operations of vector algebra is introduced with examples.



click on the content to continue..

A person takes `2` apples and then takes `3` oranges. How many fruits the person has?

  • `3`
  • `2`
  • `5`
  • `5`
  • none of the above

The answer is '`5`' fruits. In this scalar quantities `2` and `3` add to the sum `5`

Scalar quantities can be added.

A person takes `6` apples and then puts back `4` apples. How many fruits the person has?

  • `3`
  • `2`
  • `2`
  • `5`
  • none of the above

The answer is '`2`' fruits. In this scalar quantity `4` is subtracted from `6` to get result `2`.

Scalar quantities can be subtracted

A person takes `6` apples and then takes `-4` apples. How many fruits the person has?

  • `3`
  • `2`
  • `2`
  • `5`
  • none of the above

The answer is '`2`' fruits. In this, `6` is added with `-4`, which is effectively subtraction in the form `(6-4)`.

Subtraction is the inverse of addition.

A person takes `2` apples and places them in his basket. He repeats that `3` times in total. How many apples does the person have in the basket?

  • `2+2+2` apples
  • `2xx3` apples
  • `6` apples
  • all the above
  • all the above

Answer is 'all the above'. In this, scalar quantity `2` is repeatedly added `3` times to get result `6`.

A scalar quantity can be added repeatedly.
Repeated addition is a form of multiplication.

A person having `10` apples in her basket, divides that equally to two kids. How many apples will a kid have?

  • `10-:2`
  • `5`
  • count the number of twos in `10`
  • all the above
  • all the above

Answer is 'all the above'. In this, scalar quantity `10` is divided by `2`.

A scalar quantity can be divided.

A person having `10` apples in her basket, gives half of them to a kid. How many apples will the kid have?

  • `10xx1/2`
  • `10-:2`
  • `5`
  • `5`
  • all the above

Answer is 'all the above&rsquo. In this `10` is multiplied by half, which is equivalently the division `10-:2`.

Division is the inverse of multiplication.

A person takes `3` oranges that costs `10` coins each. How much the person has to pay for the fruits?

  • `40` coins
  • `30` coins
  • `30` coins
  • `10` coins
  • `50` coins

The answer is '`30`' coins. The scalar quantity `3` multiplies with another scalar quantity `10`.

Two different scalar quantities can be multiplied.

Summary

 •  Scalar quantities have magnitude measure.

 •  Scalar quantities can be added (`2` fruit + `3` fruit `= 5` fruit)

 •  A scalar quantity can be multiplied (repeated addition) (`2` apples `3` times `= 6` apples)

 •  scalar quantities can interact to form a product scalar quantity (`3` oranges at `10` coins each `= 30` coins)

Apart from the addition and multiplication,

 •  scalar quantities can be subtracted (`3` fruits `- 2` fruits) `= 1` fruit; which is inverse of addition

 •  Scalar quantities can be divided, which is inverse of multiplication.

Fundamental mathematical operations

 •  addition

 •  repeated addition (multiplication)

 •  product of quantities (multiplication)

Let us see How these are defined for vectors that has magnitude and direction.

A person walks `3` meter north and `4` meter east, what is the final distance (in meter) from the starting point?

  • `3+4`
  • `7`
  • both the above
  • `sqrt(3^2+4^2)`
  • `sqrt(3^2+4^2)`

Answer is '`sqrt(3^2+4^2)`'. In this vector quantity `3` meter north and `4` meter east are added. We used coordinate geometry to figure out the result.

Vector quantities can be added.

A person walks `3`m in a direction and continues to do the same four times. What is the final position of him from the starting point?

  • `3xx4` meter in the specified direction
  • `3+3+3+3` meter in the specified direction
  • both the above
  • both the above

Answer is 'both the above'.

Vector quantities can be repeatedly added or equivalently – vector can be multiplied by a scalar quantity.

A person pushes with 3 unit force toward east and causes 2 unit displacement towards north-east. Given that
work `=`product of force and displacement
Can you make a guess what is the work done by the person?

  • force as a vector and displacement as a vector are multiplied
  • force as a vector and displacement as a vector are multiplied
  • force as a vector is multiplied by magnitude of displacement
  • magnitude of force and magnitude of displacement are multiplied
  • none of the above

The answer is, 'force as a vector and displacement as a vector are multiplied'. Over this course, this will be explained in detail.

Vectors can be multiplied.

Vector Arithmetic involves

 •  Addition

 •  repeated addition or multiplication by scalar

 •  two types of vector products

Vector Arithmetic: Given vectors `vec p` and `vec q`,

 •  vector addition: `vec p + vec q`

 •  multiplication of vector by scalar: `vec p + vec p + cdots (n-text(times)) = n vec p`

 •  vector dot product: `vec p` multiplied by component of `vec q` in parallel to `vec p`

 •  vector cross product: `vec p` multiplied by component of `vec q` in perpendicular to `vec p`

Apart from the addition, multiplication of vector by a scalar, and vector products,

 •  vectors can be subtracted, which is 'inverse of addition' .

 •  vectors can be divided by a scalar, which is included as 'inverse of multiplication of a vector by a scalar'

 •  Vector division is not defined. We’ll see why in due course of this learning.

                            
slide-show version coming soon