| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 13 |
| 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 standard C4 vectors question covering routine techniques: finding a line equation, using parametric form to find unknowns, calculating angles with dot product, and finding perpendicular projection. All parts follow textbook methods with no novel insight required, making it slightly easier than average. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
Referred to an origin $O$, the points $A$, $B$ and $C$ have position vectors $(\mathbf{9i} - \mathbf{2j} + \mathbf{k})$, $(\mathbf{6i} + \mathbf{2j} + \mathbf{6k})$ and $(\mathbf{3i} + p\mathbf{j} + q\mathbf{k})$ respectively, where $p$ and $q$ are constants.
\begin{enumerate}[label=(\alph*)]
\item Find, in vector form, an equation of the line $l$ which passes through $A$ and $B$. [2]
\end{enumerate}
Given that $C$ lies on $l$,
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item find the value of $p$ and the value of $q$, [2]
\item calculate, in degrees, the acute angle between $OC$ and $AB$. [3]
\end{enumerate}
The point $D$ lies on $AB$ and is such that $OD$ is perpendicular to $AB$.
\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Find the position vector of $D$. [6]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 Q8 [13]}}