What's the difference between Complex Numbers and Vectors?

Question

On my assignment I'm supposed to:

1) Multiply the Complex Numbers (5∠78°)(10∠163°)
and
2) Multiply the Vectors [5,∠78°]•[10,∠163°]

I'm not sure how to go about doing either. The second one the only difference I see is the dot product and shape of the brackets. What's that have to do with anything? I'm having troubles looking up info on the first one, because everything on multiplying complex numbers comes up in the (a+b i)(c+d i) format....

Answer 1

Complex numbers and vectors in 2-dimensions both can be represented by a pair of real numbers (a,b). For complex numbers that pair represents the complex value (a + bi). For vectors it represents an "arrow" from the origin to the point (a,b).



The operations of addition and subtraction are identical for both -- you simply add the first terms together to get the first term of the result, and likewise for the second terms. Thus, in both cases,



(a,b)+(c,d) = ( a+c , b+d )



But multiplication behaves differently.



Your two numbers/vectors are given in polar form (magnitude and angle).



When we multiply two vectors using the "dot product" we get a real number (NOT another vector) that is equal to the product of the magnitudes times the cosine of the angle between them, i.e., (5)(10)cos(163-78) = 50cos(85) ~= 4.358.



When we multiply two complex numbers we multiply their magnitudes and add their angles. So the first product will be a complex number having magnitude (5)(10)=50 and angle (78+163)=241.



Complex multiplication can also expressed as:



( a + bi ) ( c + di ) = (ac-bd) + (ad+bc)i



The two forms of multiplication (polar and rectangular) are performing the same operation even though they look rather different. This is one of the many wonderful properties that make complex numbers so useful, not that vectors are any less so.

Answer 2

Note :( 5< 78° ) = 5 cos 78° + i 5 sin 78° = a + bi

Answer 3

I just want to add that C, the set of complex numbers is a field while R^2, with the usual pointwise multiplication is not a field, not even integral domain. So you can have field morphisms(functions) on C but not on R^2