| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2022 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicular distance point to line |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question involving a cuboid setup with routine calculations: finding position vectors using given ratios, writing a line equation, and computing perpendicular distance using the standard formula |OM × d|/|d|. While it requires multiple steps and careful coordinate work, all techniques are direct applications of standard methods with no novel insight required. Slightly easier than average due to the structured parts guiding the solution. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement4.04a Line equations: 2D and 3D, cartesian and vector forms |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Obtain \(\overrightarrow{OM} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}\) | B1 | |
| Use a correct method to find \(\overrightarrow{MN}\) | M1 | e.g. \(\overrightarrow{MO}+\overrightarrow{OA}+\overrightarrow{AN}\) or \(\overrightarrow{MO}+\overrightarrow{ON}\) |
| Obtain \(\overrightarrow{MN} = \mathbf{i}+\mathbf{j}-2\mathbf{k}\) | A1 | Accept any notation |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Use a correct method to form an equation for \(MN\) | M1 | Allow without \(\mathbf{r} = \ldots\) |
| Obtain \(\mathbf{r} = 2\mathbf{i}+3\mathbf{j}+\lambda(\mathbf{i}+\mathbf{j}-2\mathbf{k})\) | A1 FT | OE e.g. \(\mathbf{r}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}+\mu(\mathbf{i}+\mathbf{j}-2\mathbf{k})\). Must have \(\mathbf{r}=\ldots\) Follow their answers to part 9(a) |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\overrightarrow{OP}\) for a general point \(P\) on \(MN\) in component form, e.g. \((2+\lambda, 3+\lambda, -2\lambda)\) | B1 | |
| Equate scalar product of \(\overrightarrow{OP}\) and a direction vector for \(MN\) to zero and solve for \(\lambda\) | M1 | |
| Obtain \(\lambda = -\frac{5}{6}\) | A1 | OE e.g. \(\mu = \frac{1}{6}\) |
| Obtain \(\sqrt{\frac{53}{6}}\) correctly | A1 | AG e.g. from \(\sqrt{\left(\frac{7}{6}\right)^2+\left(\frac{13}{6}\right)^2+\left(\frac{5}{3}\right)^2}\) |
## Question 9:
### Part 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Obtain $\overrightarrow{OM} = \mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ | B1 | |
| Use a correct method to find $\overrightarrow{MN}$ | M1 | e.g. $\overrightarrow{MO}+\overrightarrow{OA}+\overrightarrow{AN}$ or $\overrightarrow{MO}+\overrightarrow{ON}$ |
| Obtain $\overrightarrow{MN} = \mathbf{i}+\mathbf{j}-2\mathbf{k}$ | A1 | Accept any notation |
### Part 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Use a correct method to form an equation for $MN$ | M1 | Allow without $\mathbf{r} = \ldots$ |
| Obtain $\mathbf{r} = 2\mathbf{i}+3\mathbf{j}+\lambda(\mathbf{i}+\mathbf{j}-2\mathbf{k})$ | A1 FT | OE e.g. $\mathbf{r}=\mathbf{i}+2\mathbf{j}+2\mathbf{k}+\mu(\mathbf{i}+\mathbf{j}-2\mathbf{k})$. Must have $\mathbf{r}=\ldots$ Follow their answers to part 9(a) |
### Part 9(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\overrightarrow{OP}$ for a general point $P$ on $MN$ in component form, e.g. $(2+\lambda, 3+\lambda, -2\lambda)$ | B1 | |
| Equate scalar product of $\overrightarrow{OP}$ and a direction vector for $MN$ to zero and solve for $\lambda$ | M1 | |
| Obtain $\lambda = -\frac{5}{6}$ | A1 | OE e.g. $\mu = \frac{1}{6}$ |
| Obtain $\sqrt{\frac{53}{6}}$ correctly | A1 | AG e.g. from $\sqrt{\left(\frac{7}{6}\right)^2+\left(\frac{13}{6}\right)^2+\left(\frac{5}{3}\right)^2}$ |
---9\\
\includegraphics[max width=\textwidth, alt={}, center]{c1fbc9ef-2dc6-43c3-bc58-179f683c9acf-16_696_1104_264_518}
In the diagram, $O A B C D E F G$ is a cuboid in which $O A = 2$ units, $O C = 4$ units and $O G = 2$ units. Unit vectors $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$ are parallel to $O A , O C$ and $O G$ respectively. The point $M$ is the midpoint of $D F$. The point $N$ on $A B$ is such that $A N = 3 N B$.
\begin{enumerate}[label=(\alph*)]
\item Express the vectors $\overrightarrow { O M }$ and $\overrightarrow { M N }$ in terms of $\mathbf { i } , \mathbf { j }$ and $\mathbf { k }$.
\item Find a vector equation for the line through $M$ and $N$.
\item Show that the length of the perpendicular from $O$ to the line through $M$ and $N$ is $\sqrt { \frac { 53 } { 6 } }$.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2022 Q9 [9]}}