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The Laser (shown in Figure 1) is a class of single-handed, one-design sailing boat.

Figure 1

https://en.wikipedia.org/wiki/Laser_(dinghy)

Figure 2

Figure 2 shows a simplified model of a Laser.

The sail is part of a plane \(p\) and the deck is part of a plane \(q\).

The mast \(AB\), contained in \(p\), is attached to the deck at \(B\) and is normal to the deck.

The boom \(CD\), contained in \(p\), is attached to the mast at \(C\).

It is given that the coordinates of \(C\) and \(D\) are \((1,0,8)\) and \((6,-12,11)\) respectively, and \(\overrightarrow{BA}=m\mathbf{k}\), where \(m\) is a positive constant.

1. Find a cartesian equation of \(p\).[3]
2. Find the exact value of \(\cos\angle ACD\).[2]
3. Sun rays perpendicular to the deck cast a shadow of the boom \(CD\) onto the deck. Find the projection vector of \(\overrightarrow{CD}\) onto \(q\).[3]

It is further given that the coordinates of \(B\) is \((1,0,2)\).

4. A gull lands on the boom at a point \(G\), which is equidistant from the mast and the deck. Find the coordinates of \(G\).[6]