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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 Dot Product

Vector Dot Product : First Principles

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

click on the content to continue..

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. 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.

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?

• |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.
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