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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.
mathsVector AlgebraVector Cross Product

### Cross Product: First Principles

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

click on the content to continue..

The product where vector components in perpendicular interact is defined. Given vectors vec p and vec q as shown in figure. 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. 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. 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. 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. 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. 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.
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
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