2022-01-17

What are the rules for complex-component vectors and why?

nick1337

Expert2022-01-19Added 699 answers

Step 1

Given complex vectors

and

their scalar (=dot) product is given by

Why? Well, you want to generalize the usual dot product on

for all v, and the vector

for all v. Most people do: the expression

should define a norm and

Having settled this, let v be a non-zero vector. Its orthogonal complement

is the set of all vectors orthogonal to v. Since U is determined by the single linear equation

it is an

which you can make into an orthonormal basis of U using Gram-Schmidt (note that the notation

is to be understood. The fact that you're working with PK\mathbb{C}\) and not with

i.e., the dot product is conjugate-linear in the second variable.

Finally, in order to solve the equation

simply take any

and put

and note that

Vasquez

Expert2022-01-19Added 597 answers

To take the length of a complex vector you need the squared magnitudes of the components. So the vector
$(1+i,\text{}1-i)$
has length
$\sqrt{|1+i{|}^{2}+|1-i{|}^{2}}=\sqrt{2+2}=2$
It is still true that the dot product of orthogonal vectors is zero in a complex space. So once you find a B parallel to A with $A\times B=d$ you can add any vector orthogonal to A to it and the dot product will not change. But when you take the dot product, remember to take the complex conjugate of the components of B.

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