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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 Cross Product

Cross Product: First Principles

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



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The product where vector components in perpendicular interact is defined. Given vectors `vec p` and `vec q` as shown in figure.vector cross product first principles What is the component of `vec q` perpendicular to the `vec p`?

  • `vec a`
  • `vec b`
  • `vec b`
  • `ai+bj`
  • `bj+ai`

The answer is '`vec b`'

Given `vec q` with component `vec b` in perpendicular to `vec p` as shown in figure.vector cross product first principles The angle between the vectors is `theta`. What is `|vec b|`?

  • `|q|sin theta`
  • `|q|sin theta`
  • `|q|cos theta`
  • `|p|sin theta`
  • `|p|cos theta`

The answer is '`|q|sin theta`'

Given vectors `vec p` and `vec q` and the product between them as shown in the figure.vector cross product first principles What will be a good choice of direction of the product between components in perpendicular?

Two intersecting lines, which are not parallel, define a plane. The normal on the face of the plane describes the plane and so, the normal is taken as the direction of the product.

A plane can be described by

 •  normal which is one side of the plane, or

 •  the negative of the normal, which is the other side of the plane. vector cross product choice of directions Which one is chosen for the vector cross product `vec p xx vec q`?

The direction of the cross product `vec p xx vec q`.vector cross product standard screw direction In a standard right-handed rectangular coordinate system,
The direction in which a standard screw advances when it turns from `vec p` to `vec q`, defines the normal.

The direction of the cross product `vec p xx vec q`.vector cross product right hand thumb rule for direction In a standard right-handed rectangular coordinate system,
The direction pointed by the right thumb when fingers are curled to point from `vec p` to `vec q`, defines the normal.

Formal definition of vector cross product. vector cross product illustration
`vec p xx vec q = |p||q|sin theta hat n` where `hat n` is the normal denoting the direction of right handed rotation from `vec p` to `vec q`.

Vector Cross Product is defined as the product of components in perpendicular.

Vector Cross Product: for vectors `vec p, vec q in bbb(R)^3`vector cross product first principles
`vec p xx vec q = (|p||q|sin theta ) hat n`
where `hat n` is the unit vector of right-handed rotation from `vec p` to `vec q`.

                            
slide-show version coming soon