I think it might be more common to represent vectors in the plane (two-dimensional vectors) in the form v = xi + yj. Here i and j are just unit vectors, having nothing to do with complex numbers. If your class is really representing vectors as complex numbers, it would be because the complex plane is isomorphic to the real plane. For example, the vector <2, 3> (or 2i + 3j) in the real plane corresponds to the complex number 2 + 3i in the complex plane.Frenemy90210 said:In mechanics, a vector is represented by complex number (x + i y). Is there a simple/intuitive explanation as to why the y component is multiplied by i , which is equal to square root of -1 ? ; In this case, did it have to be of value sqrt(-1) ? or is "i" used to keep x and y separate and not "mix" with each other ?
That is useful if rotations of vectors using complex multiplication are to be used.Frenemy90210 said:In mechanics, a vector is represented by complex number (x + i y). Is there a simple/intuitive explanation as to why the y component is multiplied by i , which is equal to square root of -1 ? ; In this case, did it have to be of value sqrt(-1) ?
Good point. That is useful in all cases. It is really x*1 + y*i, where 1 and i are the basis vectors that should never be mixed when added or subtracted. As complex numbers, they should only be related as the definition of complex multiplication specifies.or is "i" used to keep x and y separate and not "mix" with each other ?
jedishrfu said:Historically Hamilton tried to extend complex numbers into a 3D context and came up with quaternions which useda real component and i j and k imaginary components.
It had a lot going for it and was the basis for classical physics math. However some folks felt it was too complicated and developed vector methods dropping the real component and retaining the i j and k notation.
There’s been some resurgence in quaternions because they handle rotation ...
Using "i" to represent a vector is a common mathematical convention that indicates the direction of the vector. In this case, "i" stands for the unit vector in the x-direction. This allows for a standard way of representing and manipulating vectors in equations and calculations.
Yes, other letters can also be used to represent vectors, such as "j" for the unit vector in the y-direction and "k" for the unit vector in the z-direction in three-dimensional space. However, "i" is the most commonly used letter for representing a vector in the x-direction.
"i" can be thought of as a basis vector, as it is a unit vector that forms one of the basis vectors in a coordinate system. In this case, "i" represents the basis vector in the x-direction, while "j" and "k" represent the basis vectors in the y and z-directions, respectively.
The use of "i" as a basis vector has historical roots in mathematics and has become a convention in many fields of science. It allows for a consistent and standard way of representing and manipulating vectors, making mathematical equations and calculations more efficient and easier to understand.
Yes, "i" can be used to represent a vector in any coordinate system where the x-direction is defined. It is a versatile notation that can be used in Cartesian, polar, or other coordinate systems to represent the direction of a vector in relation to the x-axis.