How do I represent my 3-qubit state in the computational basis?

Question

I have taken the tensor product of $|0rangle otimes |-rangle otimes |+rangle$ which resulted in the matrix
$$begin{bmatrix}
1/2
1/2 
-1/2 
-1/2 
0 
0
0
0
end{bmatrix}.$$
How would I represent this in the computational basis basis ${|0rangle, |1rangle}$?

Adam Zalcman · Answer

The elements of your vector are coefficients of the state in the computational basis
$$
begin{array}{cc}
begin{bmatrix}
1/2
1/2 
-1/2 
-1/2 
0 
0
0
0
end{bmatrix}
&begin{matrix}
|000rangle 
|001rangle 
|010rangle 
|011rangle 
|100rangle 
|101rangle 
|110rangle 
|111rangle
end{matrix}
end{array}
$$
so
$$
|0rangleotimes|-rangleotimes|+rangle = frac{|000rangle + |001rangle - |010rangle - |011rangle}{2}.
$$
We can confirm by direct calculation in Dirac notation
$$
begin{align}
|0rangleotimes|-rangleotimes|+rangle &= |0rangleotimesfrac{|0rangle-|1rangle}{sqrt{2}} otimes frac{|0rangle+|1rangle}{sqrt{2}} 
&= frac{|0rangleotimes (|00rangle + |01rangle - |10rangle - |11rangle)}{2} 
&= frac{|000rangle + |001rangle - |010rangle - |011rangle}{2}.
end{align}
$$

How do I represent my 3-qubit state in the computational basis?

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