Real vector plus complex vector

[Neitrino]

If there are three vectors a=(1,0,0) and b=(0,1,0) what vector will be a+ib (i - is a complex)
Is it possible to imagine its direction in real three-space...

Sorry if this is basic question

 

[Phiphy]

Just add each component respectively: a+ib=(1+0i,0+i,0+0i)=(1,i,0).
You can not imagine its direction in real 3-space because complex numbers can not be ordered. But you can imagine a 6-dimensional vector with a definite direction.

 

[sir-pinski]

, I am just trying to understand.

Yes, it is possible to imagine the direction of a+ib in real three-space. To do so, we can think of a and b as representing vectors in the x-y plane, with a pointing in the positive x-direction and b pointing in the positive y-direction. When we add a+ib, we are essentially adding a complex number to a real number, which results in a vector that is a combination of both real and imaginary components.

In this case, the real component of a+ib would be (1,0,0) and the imaginary component would be (0,i,0). This means that the resulting vector would have a magnitude of 1 in the x-direction and a magnitude of i (or 0 in the y-direction).

While it may be difficult to visualize a vector with an imaginary component, we can still understand its direction in real three-space by considering the magnitude and direction of its real component. In this case, the direction of a+ib would be in the positive x-direction, with a magnitude of 1.

I hope this helps to clarify and understand the concept of adding a real vector and a complex vector.