# Orthogonal vector

## Homework Statement

Construct a third vector which is orthogonal to the following pair and normalize all three vectors:
$$\underline{a}=(1-i,1,3i), \underline{b}=(1+2i,2,1)$$

## Homework Equations

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

## The Attempt at a Solution

Mark44
Mentor
Your relevant equations are a good start, so use them.

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
Mentor
Do it the same way.

Dick
Homework Helper
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.

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
Homework Helper
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