Orthogonal vector

[Grand]

1. The problem statement, all variables and given/known data
Construct a third vector which is orthogonal to the following pair and normalize all three vectors:
\underline{a}=(1-i,1,3i), \underline{b}=(1+2i,2,1)


2. Relevant equations
\underline{c}.\underline{a}=0 and \underline{c}.\underline{b}=0 where c=(x y z)


3. The attempt at a solution

[Mark44]

Your relevant equations are a good start, so use them.

[Grand]

The question is, how do we make a dot product when the vectors are complex? Is it the same way as real vectors or not?

[Mark44]

Do it the same way.

[Dick]

Do it the same way.

Not exactly. You take the complex conjugate of the first vector before you multiply the components. Otherwise <x,x>>=0 doesn't work.

[Apphysicist]

Not exactly. You take the complex conjugate of the first vector before you multiply the components. Otherwise <x,x>>=0 doesn't work.

I'm not familiar with complex vectors, but since you want a vector that is orthogonal to both, rather than trying two dot products, wouldn't it be prudent to use a cross product?

[HallsofIvy]

That would be the method I would choose, but as Dick says, the only difference in dot product with complex components is that you use the complex conjugates of the components of one vector.

[Dick]

That would be the method I would choose, but as Dick says, the only difference in dot product with complex components is that you use the complex conjugates of the components of one vector.

I think that difference is important. I think if you take the usual definition of 'cross-product' with the complex notion of dot product, it isn't true that the cross product is orthogonal to the vectors in the product. The cross product is pretty specifically three real dimensional. C^3 isn't really three dimensional in that sense. I think you should just solve the linear equations.