| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Show lines intersect and find intersection point |
| Difficulty | Standard +0.3 This is a standard C4 vectors question requiring routine techniques: equating components to find intersection (solving simultaneous equations), then using the scalar product formula to find the angle between direction vectors. While multi-step, it involves direct application of well-practiced methods with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
Relative to a fixed origin, two lines have the equations
$$\mathbf{r} = (7\mathbf{i} - 4\mathbf{k}) + s(4\mathbf{i} - 3\mathbf{j} + \mathbf{k}),$$
and
$$\mathbf{r} = (-7\mathbf{i} + \mathbf{j} + 8\mathbf{k}) + t(-3\mathbf{i} + 2\mathbf{k}),$$
where $s$ and $t$ are scalar parameters.
\begin{enumerate}[label=(\roman*)]
\item Show that the two lines intersect and find the position vector of the point where they meet. [5]
\item Find, in degrees to 1 decimal place, the acute angle between the lines. [4]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 Q6 [9]}}