why no-one figured out vector dot and cross products : mind-boggling

orthogonality of directions

It is very surprising that some of the simple and clear patterns of scientific knowledge is not explained to the students and that makes learning very hard. This “mind-boggling” series will bring out such surprising patterns from nubtrek.com

How vector dot and cross products are explained?

Vector dot and cross products are defined in component forms and there is no connection established between them. Students are expected to remember the definition as-is.

references:

Khan Academy: Dot and Cross Product Comparison

Khan Academy: Dot Product Introduction

Khan Academy: Cross Product Introduction

How a simple and clear connection established in nubtrek?

• nubtrek first explains the orthogonality of direction: Changes in one direction does not affect other orthogonal directions.
• eg: In addition, change in one direction is limited to that direction
• Given two vector p and q, (shown in figure), q is split into two components a and b.
• a is in parallel to p
• b is in perpendicular to p

• nubtrek explains the Vector Addition, Dot Product, and Cross Product as given below
• Sum p+q = hypotenuse of right-triangle with arms p+a and b: Note that the components in parallel p and a are added.

• dot product p . q = p . a : Note that only the component in parallel to p take part in the multiplication. The other component b does not take part.

• cross product p x q = p x b : Note that only the component in perpendicular to p take part in the multiplication. The other component a does not take part.

This explanation gives a simple and clear pattern. And it is in agreement with the application scenarios too.

Having defined the dot and cross product as product of component in parallel and component in perpendicular respectively, nubtrek derives the cos or sin form and component form of the vector products.

nubtrek provides the pattern that no-other book or webpage provides. This pattern makes the whole knowledge connected and makes it very easy.

Reference from nubtrek.com:

Understanding Direction of Vectors

Dot Product: First Principles

Dot Product: Component Form

Cross Product: First Principles

Cross Product: Component Form

Thanks for reading.