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11. [Maximum mark: 18]

A line $L$ is defined by $L: -\frac{x}{2} = y + 4 = \frac{z}{3}$.

(a) Find the equation of $L$, expressing your answer in the form $\mathbf{r} = \mathbf{a} + \lambda\mathbf{b}$, where $\lambda \in \mathbb{R}$. [3]

(b) Determine the minimum distance from the origin $O$ to the line $L$. [5]

A plane $\Pi$ is defined by $\Pi: 6x - 3y + 5z = 24$.

(c) Verify that $\Pi$ contains $L$. [3]

A second line $M$ is parallel to $\Pi$.

The line $M$ passes through the point (4, 1, 2) and intersects the $z$-axis.

(d) Find the equation of $M$, expressing your answer in the form $\mathbf{s} = \mathbf{c} + \mu\mathbf{d}$, where $\mu \in \mathbb{R}$. [7]