Standard +0.3 This is a straightforward 3D vectors question requiring coordinate setup, vector arithmetic, and dot product application. While it involves multiple steps (finding positions, computing vectors, dot product, then angle), each step uses standard techniques with no conceptual challenges. The geometric setup is clearly described, making this slightly easier than average.
\includegraphics{figure_7}
The diagram shows a solid cylinder standing on a horizontal circular base with centre \(O\) and radius \(4\) units. Points \(A\), \(B\) and \(C\) lie on the circumference of the base such that \(AB\) is a diameter and angle \(BOC = 90°\). Points \(P\), \(Q\) and \(R\) lie on the upper surface of the cylinder vertically above \(A\), \(B\) and \(C\) respectively. The height of the cylinder is \(12\) units. The mid-point of \(CR\) is \(M\) and \(N\) lies on \(BQ\) with \(BN = 4\) units.
Unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are parallel to \(OB\) and \(OC\) respectively and the unit vector \(\mathbf{k}\) is vertically upwards.
Evaluate \(\overrightarrow{PN} \cdot \overrightarrow{PM}\) and hence find angle \(MPN\). [7]
\includegraphics{figure_7}
The diagram shows a solid cylinder standing on a horizontal circular base with centre $O$ and radius $4$ units. Points $A$, $B$ and $C$ lie on the circumference of the base such that $AB$ is a diameter and angle $BOC = 90°$. Points $P$, $Q$ and $R$ lie on the upper surface of the cylinder vertically above $A$, $B$ and $C$ respectively. The height of the cylinder is $12$ units. The mid-point of $CR$ is $M$ and $N$ lies on $BQ$ with $BN = 4$ units.
Unit vectors $\mathbf{i}$ and $\mathbf{j}$ are parallel to $OB$ and $OC$ respectively and the unit vector $\mathbf{k}$ is vertically upwards.
Evaluate $\overrightarrow{PN} \cdot \overrightarrow{PM}$ and hence find angle $MPN$. [7]
\hfill \mbox{\textit{CAIE P1 2018 Q7 [7]}}