| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from origin to line |
| Difficulty | Standard +0.3 This is a structured multi-part question covering standard C4 vector techniques: finding a displacement vector, using the scalar product for angles and perpendicularity, and verifying a point lies on a line. All parts follow routine procedures with clear signposting, though part (d) requires combining concepts. Slightly easier than average due to the scaffolded structure and straightforward calculations. |
| Spec | 1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors4.04c Scalar product: calculate and use for angles |
Relative to a fixed origin $O$, the point $A$ has position vector $4\mathbf{i} + 8\mathbf{j} - \mathbf{k}$, and the point $B$ has position vector $7\mathbf{i} + 14\mathbf{j} + 5\mathbf{k}$.
\begin{enumerate}[label=(\alph*)]
\item Find the vector $\overrightarrow{AB}$. [1]
\item Calculate the cosine of $\angle OAB$. [3]
\item Show that, for all values of $\lambda$, the point $P$ with position vector
$$\lambda\mathbf{i} + 2\lambda\mathbf{j} + (2\lambda - 9)\mathbf{k}$$
lies on the line through $A$ and $B$. [3]
\item Find the value of $\lambda$ for which $OP$ is perpendicular to $AB$. [3]
\item Hence find the coordinates of the foot of the perpendicular from $O$ to $AB$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q13 [12]}}