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summary of this topic

Vector Dot Product

Vector Dot Product

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Vector Dot Product: First Principles


 »  multiplied with component in parallel
    →  `vec p cdot vec q ` ` =vec p cdot vec a` `= |vec p||vec a|`
    →  product of magnitudes of components in parallel

Vector Dot Product : First Principles

plain and simple summary

nub

plain and simple summary

nub

dummy

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

simple steps to build the foundation

trek

simple steps to build the foundation

trek

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


Keep tapping on the content to continue learning.
Starting on learning "the first principles of vector dot product". ;; In this page, the vector dot product is explained in first principles.

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`
  • 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|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.

comprehensive information for quick review

Jogger

comprehensive information for quick review

Jogger

dummy

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`.



           

practice questions to master the knowledge

Exercise

practice questions to master the knowledge

Exercise

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

The answer is 'All the above'

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

  • `0`
  • `24`
  • `-24`
  • `3/2`

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

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`
  • `-600`
  • `-300 sqrt 2`

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

your progress details

Progress

About you

Progress

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 s r is split into components s a and s b, where s a is in the same direction as the cause f 1. What is the work done by cause f 1?
f;1;muliplied;s;a
f 1 multiplied by s a
cannot;calculate
cannot calculate
The 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 p and 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 p dot vector q = magnitude of p multiplied magnitude of a where magnitude of a is the component of vector q in the direction of vector p.
How can the component a be computed?
sine
q sine theta
cos;cause
q cos theta
tan
q tan theta
magnitude
magnitude of q
The answer is "q cos theta", using basic geometry.
This product is formally defined as vector dot product. ; vector p dot vector q = magnitude of p multiplied by magnitude of q multiplied by 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 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 or Scalar Product is defined as the product of components in parallel (i.e. in the same direction).
Vector Dot Product or Scalar Product is defined as vector p dot vector q = magnitude of p multiplied by magnitude of q multiplied by cos theta ;; where theta is the angle between vector p and vector q.
Two vectors p and q with magnitudes 2 and 3 respectively are at an angle 60 degree. What is vector p dot vector q?
1
2
3
4
The answer is 'All the above'
Vector with magnitudes 4 and 6 are in the opposite directions. What is the dot product of these two vectors?
1
2
3
4
The answer is "minus 24". The angle between the vectors is 180 degree and cos 180 equals minus 1.
A vector p of magnitude 30 is the cause. The observed effect is q of magnitude 20 at an angle 45 degree to the vector p. What is the product between them?
1
2
3
4
The answer is "300 square root 2". Product between cause and effect is the vector dot product magnitude of p, magnitude of q, cos theta.

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