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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 AlgebraVector Dot Product

Vector Dot Product : First Principles

In this page, the vector dot product is explained in first principles.



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We have developed the understanding that
 •  Effect will be in the same direction as that of the cause.
 •  the observed effect can be sum of the effects of multiple different causes.

A vector can be split into components in parallel and in perpendicular directions.

The effect `vec s_r` is split into components `vec s_a` and `vec s_b`, where `vec s_a` is in the same direction as the cause `vec f_1`. vector dot product illustration What is the work done by cause `f_1`?

  • `f_1` multiplied by `s_a`
  • `f_1` multiplied by `s_a`
  • cannot calculate

Answer is '`f_1` multiplied by `s_a`' as the effect is in the same direction and in that direction no other cause is observed.

Generalizing this... For two vectors `vec p` and `vec q`, the vector dot product of the vectors is given by multiplication of
 •  magnitude of one vector and
 •  the magnitude of the component of second vector in parallel to the first vector.

vector dot product illustration
`vec p cdot vec q = |p| |a|`
where `|a|` is the component of `vec q` in the direction of `vec p`.

How can the component `a` be computed? vector dot product illustration

  • `|q|sin theta`
  • `|q|cos theta`
  • `|q|cos theta`
  • `|q|tan theta`
  • `|q|`

Answer is '`|q|cos theta`', using basic trigonometry.

This product is formally defined as vector dot product. vector dot product illustration
`vec p cdot vec q = |p||q|cos theta`
Note that the result is a scalar. So, this operation is also called Scalar Product of vectors.

A vector multiplication for components in parallel (in the same direction) is defined as scalar product or dot product.

The components in perpendicular do not take part in the dot product as the cause-effect pairs are always in the same direction. For a cause vector, an effect vector in orthogonal (at an angle `90^@`) to the cause is not possible.

Though the vector dot product is explained for cause-effect pairs, the product can be defined as product between vectors which are not cause-effect pair but interact to form a product. In such cases, if the components in parallel interact to form the product, the vector dot product models the interaction.

The type of interaction is explained with an example in the chapter vector-cross-products.

Vector Dot Product / Scalar Product is defined as the product of components in parallel (i.e. in the same direction).

Vector Dot Product / Scalar Product : is defined as
`vec p cdot vec q = |p||q|cos theta` where `theta` is the angle between `vec p` and `vec q`.

Solved Exercise Problem:

Two vectors `vec p` and `vec q` with magnitudes `2` and `3` respectively are at an angle `60^@`. What is `vec p cdot vec q`?

  • `2 xx 3 xx cos 60^@`
  • `2 xx 3 xx 1/2`
  • `3`
  • all the above
  • all the above

The answer is 'All the above'

Solved Exercise Problem:

Vector with magnitudes `4` and `6` are in the opposite directions. What is the dot product of these two vectors?

  • `0`
  • `24`
  • `-24`
  • `-24`
  • `3/2`

The answer is '`-24`'. The angle between the vectors is `180^@` and `cos180^@ = -1`.

Solved Exercise Problem:

A vector `vec p` of magnitude `30` is the cause. The observed effect is `vec q` of magnitude `20` at an angle `45^@` to the `vec p`. What is the product between them?

  • `600`
  • `300 sqrt 2`
  • `300 sqrt 2`
  • `-600`
  • `-300 sqrt 2`

The answer is '`300 sqrt 2`'. Product between cause and effect is vector dot product `|p||q|cos theta`.

                            
slide-show version coming soon