2025 TJC Promo Q3

With reference to the origin \(O\), the points \(A\), \(B\) and \(X\) have position vectors \(3\mathbf{i}+2\mathbf{j}+\mathbf{k}\), \(\mathbf{i}+2\mathbf{j}+3\mathbf{k}\) and \(11\mathbf{i}+10\mathbf{j}+9\mathbf{k}\) respectively. The point \(C\) lies on \(OA\) produced such that \(OA:AC=1:\alpha\), where \(\alpha\) is a constant. The point \(D\) lies on \(OB\) produced such that \(OB:BD=1:\beta\), where \(\beta\) is a constant.

It is given that \(OCXD\) is a parallelogram.

1. Write down the position vectors of \(C\) and \(D\) in terms of \(\alpha\) and \(\beta\).[1]
2. Show that \(\beta=1\) and find the value of \(\alpha\).[3]
3. Find the exact shortest distance from \(X\) to line \(OD\).[3]