| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | 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 vector operations (finding vectors between points, dot products for angles and perpendicularity, parametric line equations). Each part builds systematically on the previous one with clear signposting. While it requires multiple techniques, all are routine C4 vector methods with no novel problem-solving insight needed. Slightly easier than average due to its scaffolded structure. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
I don't see any extracted mark scheme content in your message. You've provided the question number "Question 4: 4" but no actual mark scheme content to clean up.
Please provide the mark scheme text that needs to be converted to LaTeX notation and formatted.4. 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 $\overrightarrow { A B }$.
\item Calculate the cosine of $\angle O A B$.
\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 $B$.
\item Find the value of $\lambda$ for which $O P$ is perpendicular to $A B$.
\item Hence find the coordinates of the foot of the perpendicular from $O$ to $A B$.
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q4 [12]}}