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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

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Multiplication of Vectors

» Two products because of orthogonality of components of vectors

→ dot product is defined for components in parallel `vec p cdot vec q` `=vec p cdot vec a`

→ cross product is defined for components in perpendicular `vec p xx vec q` `=vec p xx vec b`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

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.

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trek

*simple steps to build the foundation*

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In this page, learn in detail about basics of 'vector cross product' with an example.

Starting on learning "Introduction to Cross Product". ;; In this page, learn in detail about basics of 'vector cross product' with an example.

We have learned that Vector dot product is defined as multiplication of components in parallel. This definition provides mathematical model for the cause and effect which are in the same direction.

In this page, vector cross product is introduced step-by-step.

A person takes `3` oranges that costs `10` coins each. How much the person has to pay for the fruits?

- `40` coins
- `30` coins
- `10` coins
- `50` coins

The answer is '`30`' coins. The scalar quantity `3` multiplies with another scalar quantity `10`. *Two different scalar quantities can be multiplied.*

That is, two different quantities are multiplied. One quantity is independent of another.

In the case of vectors too, two quantities can interact. The quantities need not be related and are independent of one another.

**Effect of direction : ** In mathematical calculations, vectors have the following properties

• A vector is represented as components along orthogonal directions.

• In vector addition, components in parallel add up.

• Two types of vector multiplication are defined for component in parallel and component in perpendicular.

The vector components in parallel form a product called dot product. Vector dot product has meaning or practical application -- cause and effect are in parallel.

Similar to that, does vector cross product has any meaning or practical application? OR is that just an abstraction?

Consider the field of standing crops and a blade is cutting the crop at an angle. Cross product (in which components in perpendicular interact) can be understood with the following

• If the blade runs at right angle to the crop, then it cuts maximum

• If the blade cuts vertically in parallel to the crop, it does not cut any crop

• If the blade cuts at an angle `theta`, then the crop cut is in proportion to the component in perpendicular to the crop

And, of course, the cross product is also used in the abstract form to compute products between vectors.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

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`

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

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We have learned that Vector dot product is defined as multiplication of components in parallel. This definition provides mathematical model for the cause and effect which are in the same direction. ;;In this page, vector cross product is introduced step-by-step.

A person takes 3 oranges that costs 10 coins each. How much the person has to pay for the fruits?

40

40 coins

30

30 coins

10

10 coins

50

50 coins

The answer is "30 coins". The scalar quantity 3 multiplied with another scalar quantity 10. ; Two different scalar quantities can be multiplied.

That is, two different quantities are multiplied. One quantity is independent of another. In the case of vectors too, two quantities can interact. The quantities need not be related and are independent of one another.

Effect of direction: In mathematical calculations, vectors have the following properties ;; A vector is represented as components along orthogonal directions.;; In vector addition, components in parallel add up.;; Two types of vector multiplication are defined for component in parallel and component in perpendicular.

The vector components in parallel form a product called dot product. Vector dot product has meaning or practical application -- cause and effect are in parallel. ;; Similar to that, does vector cross product has any meaning or practical application? OR is that just an abstraction? ;; Consider the field of standing crops and a blade is cutting the crop at an angle. Cross product (in which components in perpendicular interact); can be understood with the following ;; If the blade runs at right angle to the crop, then it cuts maximum ;; If the blade cuts vertically in parallel to the crop, it does not cut any crop ;; If the blade cuts at an angle theta, then the crop cut is in proportion to the component in perpendicular to the crop ; And, of course, the cross product is also used in the abstract form to compute products between vectors.

When two vector quantities interact to form a product, either one of the ;; component in parallel or ;; 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 vector p and multiplier vector q. Vector q is split into vector a and vector b, such that ;; vector q = vector a + vector b ;; vector a is in parallel to vector p; vector b is in perpendicular to vector 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. vector p dot vector q = p multiplied a ;; another with component in perpendicular to the vector, called vector cross product. vector p cross vector q = p multiplied b as vectors.