2020 TJC P1 Q9

A camper built a tripod tent using \(3\) poles and \(2\) canvases as shown in the diagram below. The \(3\) poles rest on sloping ground at the points \(A\), \(B\), and \(C\), and are fastened together at the point \(V\). With respect to the origin (not shown in the diagram), \(A\), \(B\), and \(C\) have position vectors and \(\mathrm{2}\mathbf{i}\mathrm{+3}\mathbf{j}-\mathbf{k}\), \(\mathrm{7}\mathbf{i}\mathrm{+4}\mathbf{j}\) and \(8\mathbf{i}+12\mathbf{j}+2\mathbf{k}\) respectively, where \(\mathbf{k}\) is perpendicular to the horizontal. You may assume that the sloping ground is a plane.

1. Find a vector equation, in parametric form, of the sloping ground.[2]
2. The vectors \(\overrightarrow{AV}\) and \(\overrightarrow{CV}\) are parallel to \(4\mathbf{i}+\alpha \mathbf{j}+10\mathbf{k}\) and \(-2\mathbf{i}-2\mathbf{j}+10\mathbf{k}\) respectively. Find, in either order, the value of \(\alpha \) and the position vector of \(V\).[4]
3. Find the acute angle between the pole \(CV\) and the sloping ground.[3]
4. The camper attaches a small torchlight to the top of the tent at \(V\). The torchlight was not fastened properly and dropped vertically down to the ground. Assuming the lowest end of the torch is at \(V\), find the distance travelled by the torchlight from \(V\) to the ground.[3]