| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Marks | 13 |
| Paper | 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 with routine procedures: finding a line equation (2 marks), using collinearity to find constants (2 marks), calculating an angle between vectors (3 marks), and finding a perpendicular foot (6 marks). All parts use well-practiced techniques with no novel insight required, making it slightly easier than average for A-level. |
| Spec | 4.04a 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 $(9\mathbf{i} - 2\mathbf{j} + \mathbf{k})$, $(6\mathbf{i} + 2\mathbf{j} + 6\mathbf{k})$ and $(3\mathbf{i} + p\mathbf{j} + q\mathbf{k})$ 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*)]
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\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 Q24 [13]}}