Why use i to represent y vector ?

[Frenemy90210]

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 ?

 

[scottdave]

When representing complex numbers geometrically, the imaginary component points 90° from the real component. So if the x direction represents real, then the y direction represents imaginary. There are some neat things you can do with 2-D vectors, when representing them as complex numbers.

 

[Mark44]

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 ?

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.

 

[anorlunda]

OP, you may find this PF Insights article interesting. It talks about the evolution of thought and notation for complex numbers.

https://www.physicsforums.com/insights/case-learning-complex-math/

 

[jedishrfu]

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 aspects that vectors lost when they dropped the real component.

Wikipedia has an article on the history of quaternions.

https://en.m.wikipedia.org/wiki/Quaternion

 

[FactChecker]

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) ?

That is useful if rotations of vectors using complex multiplication are to be used.

or is "i" used to keep x and y separate and not "mix" with each other ?

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.

 

[scottdave]

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 ...

Here is a nice Numberphile video about quaternions.