Hey there!
We have that \(OX = \vec{x}, OY = \vec{y}, OZ = \vec{z}\) and we can also use part (i) to conclude that \(OM = \frac{1}{2}\vec{x} + \frac{1}{2}\vec{z}\) (if you need clarification on why this is the case, please ask again!).
We can also deduce that \(XY = \vec{y} - \vec{x}\) and that from the fact that \(XY:YP = 2:3\), we also have that \(YP = \frac{3}{2}(\vec{y} - \vec{x})\). Considering that \(OY + YP = OP\), we have that \(OP = \frac{5}{2}\vec{y}-\frac{3}{2}\vec{x}\).
Now, we have a triangle with points \(O, M\) and \(P\). Try applying the result found in part (i) to find\(S\) in terms of \(\vec{y}\) and \(\vec{z}\). Hints (but not answer - it is a proof) provided below, hints are progressive and try to use as few as possible! (none if you can)
Hint 1
Express \(S\) in the form given in part (i) in terms of \(OM\) and \(OP\).
Hint 2
What does \(S\) being on \(YZ\) tell you?
Hint 3
If \(S\) is on \(YZ\), then \(S\) is necessarily independent of \(\vec{x}\). ie. Find a value for \(\lambda\) such that \(\alpha \vec{x} = 0\) for some constant \(\alpha\).
Hint 4
\(\lambda = \frac{3}{4}\) - rearrange to get the result given in the question.
Hope this helps :)