4 Relative to an origin \(O\), the position vectors of points \(P\) and \(Q\) are given by
$$\overrightarrow { O P } = \left( \begin{array} { r }
- 2
3
1
\end{array} \right) \quad \text { and } \quad \overrightarrow { O Q } = \left( \begin{array} { l }
2
1
q
\end{array} \right)$$
where \(q\) is a constant.
- In the case where \(q = 3\), use a scalar product to show that \(\cos P O Q = \frac { 1 } { 7 }\).
- Find the values of \(q\) for which the length of \(\overrightarrow { P Q }\) is 6 units.
\includegraphics[max width=\textwidth, alt={}]{933cdfe1-27bb-450d-8b9a-b494916242cb-3_647_741_845_699}
The diagram shows the cross-section of a hollow cone and a circular cylinder. The cone has radius 6 cm and height 12 cm , and the cylinder has radius \(r \mathrm {~cm}\) and height \(h \mathrm {~cm}\). The cylinder just fits inside the cone with all of its upper edge touching the surface of the cone.