What happens to multi-qubit quantum state after one of qubits is measured?
The amplitudes are scaled so that their norm is 1. In your case, if the original amplitudes are $\alpha_{000},\ldots,\alpha_{111}$, then after measuring the second bit to be 1, which happens with probability $p = \alpha_{010}^2 + \alpha_{011}^2 + \alpha_{110}^2 + \alpha_{111}^2$, the quantum state is
$$
\frac{\alpha_{010} \left|010\right\rangle + \alpha_{011} \left|011\right\rangle + \alpha_{110} \left|110\right\rangle + \alpha_{111} \left|111\right\rangle}{\sqrt{\alpha_{010}^2 + \alpha_{011}^2 + \alpha_{110}^2 + \alpha_{111}^2}}.
$$