Suppose we measure an observable $A=\hat{n} \cdot \vec{\sigma}$on a qubit, where $\hat{n}=\left(n_{x}, n_{y}, n_{z}\right) \in \mathbb{R}^{3}$ is a unit vector and $\vec{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right)$ is the vector of Pauli operators.

(i) Express $A$ as a $2 \times 2$ matrix in terms of the components of $\hat{n}$.

(ii) Representing $\hat{n}$ in terms of spherical polar coordinates as $\hat{n}=(1, \theta, \phi)$, rewrite the above matrix in terms of the angles $\theta$ and $\phi$.

(iii) What are the possible outcomes of the above measurement?

(iv) Suppose the qubit is initially in a state $|1\rangle$. What is the probability of getting an outcome 1?

(v) Consider the three-qubit state

$$|\psi\rangle=a|000\rangle+b|010\rangle+c|110\rangle+d|111\rangle+e|100\rangle$$

Suppose the second qubit is measured relative to the computational basis. What is the probability of getting an outcome 1? State the rule that you are using.