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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

### Multiplication of Vectors

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What are the fundamental operations of mathematics?

• Multiplication and Division
• all the above
• all the above

The answer is 'All the Above'. The four fundamental operations of mathematics are Addition, Subtraction, Multiplication, and Division.

We have studied about vector addition and multiplication of vectors by a scalar.

Is there something as vector multiplication?
That is the subject of learning in these pages. Let us go through some basics to understand vector multiplication.

Consider the vec p and vec q shown in figure. vec q is split into vec a and vec b, such that
vec q = vec a + vec b and
vec a is in parallel to vec p
vec b is in perpendicular to vec p
When the vectors vec p and vec q interact as a product which component(s) of vec q take part in the multiplication?

• only vec a
• only vec b
• either one : vec a or vec b
• either one : vec a or vec b
• both together : vec a and vec b

The answer is 'either one : vec a or vec b'. Remember the orthogonality of directions. When vectors interact to form products the interaction is either with the component in parallel or the component in perpendicular.

Orthogonality of Directions : Any change in a direction affects component along that direction only and does not affect the components in the directions at 90^@ to that direction.

The component which is in perpendicular to a direction does not affect the component in the direction. The directions which are at 90^@ angle are referred to as orthogonal directions.

In arithmetic operations like addition, subtraction, multiplication by a scalar, and product of vectors, – the results are computed for

•  components in parallel

•  components in perpendicular

Each of these two components interact differently.

When two vector quantities interact to form a product, either one of the (1) component in parallel or (2) component in perpendicular is involved in the multiplication. In practical scenarios, when one component interacts, the other component does not interact and is lost in the product.

How does direction affect vector quantities in multiplication? At an abstract level, there are two products possible. Given multiplicand vec p and multiplier vec q. vec q is split into vec a and vec b, such that
vec q = vec a + vec b and
vec a is in parallel to vec p
vec b is in perpendicular to vec p

two forms of multiplications are defined for each of these two components.

•  one with component in parallel to the other, called vector dot product.vec p cdot vec q = vec p cdot vec a

•  another with component in perpendicular to the vector, called vector cross product. vec p times vec q = vec p times vec b

slide-show version coming soon