| Exam Board | SPS |
|---|---|
| Module | SPS FM Pure (SPS FM Pure) |
| Year | 2025 |
| Session | June |
| Marks | 5 |
| Topic | Vectors 3D & Lines |
| Type | Angle with unknown parameter |
| Difficulty | Challenging +1.8 This is a Further Maths vector problem requiring students to set up and solve a system using dot product formulas with the double angle constraint. It demands careful algebraic manipulation of cos(2θ) = 2cos²(θ) - 1 combined with perpendicularity conditions, going beyond routine dot product exercises but following a clear logical path once the setup is recognized. |
| Spec | 4.04c Scalar product: calculate and use for angles |
The three dimensional non-zero vector $\mathbf{u}$ has the following properties:
\begin{itemize}
\item The angle $\theta$ between $\mathbf{u}$ and the vector $\begin{pmatrix} 1 \\ 5 \\ 9 \end{pmatrix}$ is acute.
\item The (non-reflex) angle between $\mathbf{u}$ and the vector $\begin{pmatrix} 9 \\ 5 \\ 1 \end{pmatrix}$ is $2\theta$.
\item $\mathbf{u}$ is perpendicular to the vector $\begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$.
\end{itemize}
Find the angle $\theta$.
[5]
\hfill \mbox{\textit{SPS SPS FM Pure 2025 Q14 [5]}}