| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | 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 testing routine techniques: dot product for perpendicularity, solving simultaneous equations for intersection, and substitution to check if a point lies on a line. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.10f Distance between points: using position vectors4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting4.04f Line-plane intersection: find point |
Two submarines are travelling in straight lines through the ocean. Relative to a fixed origin, the vector equations of the two lines, $l_1$ and $l_2$, along which they travel are
$$\mathbf{r} = 3\mathbf{i} + 4\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k})$$
and $\mathbf{r} = 9\mathbf{i} + \mathbf{j} - 2\mathbf{k} + \mu (4\mathbf{i} + \mathbf{j} - \mathbf{k})$,
where $\lambda$ and $\mu$ are scalars.
\begin{enumerate}[label=(\alph*)]
\item Show that the submarines are moving in perpendicular directions. [2]
\item Given that $l_1$ and $l_2$ intersect at the point $A$, find the position vector of $A$. [5]
\end{enumerate}
The point $B$ has position vector $10\mathbf{j} - 11\mathbf{k}$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Show that only one of the submarines passes through the point $B$. [3]
\item Given that 1 unit on each coordinate axis represents 100 m, find, in km, the distance $AB$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q6 [12]}}