Answer: (a) \(\frac{4\mathbf{i}+\mathbf{j}}{\sqrt{17}}\), (b) \(\lambda=-11\).

Answer: (a) \(\frac{4\mathbf{i}+\mathbf{j}}{\sqrt{17}}\), (b) \(\lambda=-11\).

(a) First find \(\overrightarrow{PQ}\):

Since \(\overrightarrow{PQ}=5\overrightarrow{PR}\),

\(\overrightarrow{PR}=\mathbf{i}+3\mathbf{j}.\)

Therefore

The magnitude of \(\overrightarrow{OR}\) is

\(\sqrt{4^2+1^2}=\sqrt{17}.\)

So the unit vector in the direction \(\overrightarrow{OR}\) is

\(\frac{4\mathbf{i}+\mathbf{j}}{\sqrt{17}}.\)

(b) Since \(S\) has position vector \(\lambda\mathbf{j}\), its coordinates are \((0,\lambda)\). From part (a), \(R=(4,1)\).

Hence

\(\overrightarrow{RS}=(-4)\mathbf{i}+(\lambda-1)\mathbf{j}.\)

For \(RS\) to be parallel to \(PQ\), the vector \((-4,\lambda-1)\) must be parallel to \((5,15)\).

Thus

\(\frac{\lambda-1}{-4}=\frac{15}{5}=3.\)

So

\(\lambda-1=-12,\)

giving

\(\lambda=-11.\)