- #1
parshyaa
- 307
- 19
Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.
parshyaa said:Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.
I tend to see it the otherway around. It is not an invention, it simply is there. You first deal with vectors and the diagram comes next, when you try to explain it to someone else. It's not really needed. And the same holds for the cross-product. It is simply some determinants which happen to represent a size (length, area, volume). E.g. the orthogonality is forced by definition: ##\begin{bmatrix} * \\ * \\ 0 \end{bmatrix} \times \begin{bmatrix} * \\ * \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ * \end{bmatrix}.##parshyaa said:I just want to know how it came to their mind, there is a idea for every thing , just like we know that A + B = C (A,B,C are vectors) and this can be represented by the diagram given below
That's the point! If someone asks you ... then you draw a diagram ... In case of the cross-product it turns out to be the right hand rule that often appears in nature.If someone ask, why you represented addition of vector like given in above diagram.
No. It exists without a diagram, a right hand rule, an area or whatever. It is just as useful as complex numbers are useful without having a complex plane, or integers are without being an accountant or bookmaker.Then there is a answer ... therefore I think that there may be a good reason for the cross product question
For me its a cool invention , but I got your point.fresh_42 said:I tend to see it the otherway around. It is not an invention, it simply is there. You first deal with vectors and the diagram comes next, when you try to explain it to someone else. It's not really needed. And the same holds for the cross-product. It is simply some determinants which happen to represent a size (length, area, volume). E.g. the orthogonality is forced by definition: ##\begin{bmatrix} * \\ * \\ 0 \end{bmatrix} \times \begin{bmatrix} * \\ * \\ 0 \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ * \end{bmatrix}.##That's the point! If someone asks you ... then you draw a diagram ... In case of the cross-product it turns out to be the right hand rule that often appears in nature.No. It exists without a diagram, a right hand rule, an area or whatever. It is just as useful as complex numbers are useful without having a complex plane, or integers are without being an accountant or bookmaker.
A difference on the philosophical point of view. Sometimes as well the difference between more "applied" and more "pure" scientists.parshyaa said:For me its a cool invention , but I got your point.
I think of it this way. We have a different product, the dot product, which takes a pair of vectors and gives a scalar. So we want the cross product to give a vector.parshyaa said:Why mathematicians defined that the cross product of vector A and B will be a vector perpendicular to them.
The vector cross product is a mathematical operation that takes two vectors as inputs and produces a third vector that is perpendicular to both of the input vectors. It is denoted by the symbol "×" and is also known as the vector product or the cross product.
The vector cross product is calculated using the following formula:
A × B = |A||B|sinθ
Where A and B are the two input vectors, |A| and |B| are their magnitudes, and θ is the angle between them.
The perpendicular result in the vector cross product is significant because it represents a new vector that is perpendicular to both of the input vectors. This means that the resulting vector is at a 90-degree angle from both input vectors, creating a new direction in space.
The direction of the resulting vector in the vector cross product is determined by the right-hand rule. This rule states that if the fingers of your right hand curl in the direction of the first vector, and then curl towards the second vector, your thumb will point in the direction of the resulting vector.
The vector cross product has many real-world applications, including:
- Calculating torque in physics and engineering
- Determining the direction of magnetic fields
- Creating 3D graphics and computer animations
- Navigation and orientation in robotics and autonomous vehicles
- Studying fluid dynamics and aerodynamics in engineering and meteorology