Introduction to Cross Product
In this page, learn in detail about basics of 'vector cross product' with an example.
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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` coins
- `30` 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.
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`