| Exam Board | Edexcel |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2013 |
| Session | June |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from origin to line |
| Difficulty | Moderate -0.3 This is a standard C4 vectors question testing routine techniques: substituting into a line equation, finding angles using dot products, writing parallel line equations, calculating magnitudes, and finding perpendicular distances. All parts follow textbook methods with no novel problem-solving required, making it slightly easier than average but still requiring multiple vector techniques. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication4.04c Scalar product: calculate and use for angles |
With respect to a fixed origin $O$, the line $l_1$ has vector equation
$$\mathbf{r} = \begin{pmatrix} -9 \\ 8 \\ 5 \end{pmatrix} + \mu \begin{pmatrix} 5 \\ -4 \\ -3 \end{pmatrix}$$
where $\mu$ is a scalar parameter.
The point $A$ is on $l_1$ where $\mu = 2$.
\begin{enumerate}[label=(\alph*)]
\item Write down the coordinates of $A$.
[1]
The acute angle between $OA$ and $l_1$ is $\theta$, where $O$ is the origin.
\item Find the value of $\cos \theta$.
[3]
The point $B$ is such that $\overrightarrow{OB} = 3\overrightarrow{OA}$.
The line $l_2$ passes through the point $B$ and is parallel to the line $l_1$.
\item Find a vector equation of $l_2$.
[2]
\item Find the length of $OB$, giving your answer as a simplified surd.
[1]
The point $X$ lies on $l_2$. Given that the vector $\overrightarrow{OX}$ is perpendicular to $l_2$,
\item find the length of $OX$, giving your answer to 3 significant figures.
[3]
\end{enumerate}
\hfill \mbox{\textit{Edexcel C4 2013 Q4 [10]}}