| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2021 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | 3D geometry applications |
| Difficulty | Standard +0.8 This is a multi-part 3D vector geometry question requiring: (a) vector arithmetic with parallel sides condition, (b) standard line equation recall, and (c) distance from point to line in 3D plus trapezium area calculation. Part (c) requires understanding perpendicular distance between parallel lines and applying the cross product or projection method, which goes beyond routine exercises. The combination of concepts and the non-trivial distance calculation elevate this above average difficulty. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State or imply \(\overrightarrow{AB} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k}\) | B1 | OE |
| Carry out a correct method to find \(\overrightarrow{OD}\) | M1 | |
| Obtain answer \(-4\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) | A1 | OE |
| Total | 3 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| State \(\mathbf{r} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})\) | B1FT | OE. The FT is on \(\overrightarrow{AB}\) |
| Total | 1 |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| For a general point \(P\) on \(AB\), state \(\overrightarrow{CP}\) or \(\overrightarrow{DP}\) in component form, e.g. \(\overrightarrow{CP} = (3-2\lambda,\ -\lambda,\ -6+2\lambda)\) | *M1 | |
| Equate a relevant scalar product to zero *or* equate derivative of \(\ | \overrightarrow{CP}\ | \) to zero *or* use Pythagoras in a relevant triangle and solve for \(\lambda\) |
| Obtain \(\lambda = 2\) | A1 | |
| Show the perpendicular is of length 3 | A1 | |
| Carry out a correct method to find the area of \(ABCD\) and obtain the answer 18 | A1 | |
| Alternative method: | ||
| Use a scalar product to find the projection \(CN\) (or \(DN\)) of \(BC\) (or \(AD\)) on \(CD\) | *M1 | |
| Obtain \(CN = 3\) (or \(DN = 3\)) | A1 | |
| Use Pythagoras to obtain \(BN\) (or \(AN\)) | DM1 | |
| Obtain answer 3 | A1 | |
| Carry out a correct method to find the area of \(ABCD\) and obtain the answer 18 | A1 | |
| Total | 5 |
## Question 9(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| State or imply $\overrightarrow{AB} = 2\mathbf{i} + \mathbf{j} - 2\mathbf{k}$ | B1 | OE |
| Carry out a correct method to find $\overrightarrow{OD}$ | M1 | |
| Obtain answer $-4\mathbf{i} - \mathbf{j} + 3\mathbf{k}$ | A1 | OE |
| **Total** | **3** | |
## Question 9(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| State $\mathbf{r} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} + \lambda(2\mathbf{i} + \mathbf{j} - 2\mathbf{k})$ | B1FT | OE. The FT is on $\overrightarrow{AB}$ |
| **Total** | **1** | |
## Question 9(c):
| Answer | Mark | Guidance |
|--------|------|----------|
| For a general point $P$ on $AB$, state $\overrightarrow{CP}$ or $\overrightarrow{DP}$ in component form, e.g. $\overrightarrow{CP} = (3-2\lambda,\ -\lambda,\ -6+2\lambda)$ | *M1 | |
| Equate a relevant scalar product to zero *or* equate derivative of $\|\overrightarrow{CP}\|$ to zero *or* use Pythagoras in a relevant triangle and solve for $\lambda$ | DM1 | |
| Obtain $\lambda = 2$ | A1 | |
| Show the perpendicular is of length 3 | A1 | |
| Carry out a correct method to find the area of $ABCD$ and obtain the answer 18 | A1 | |
| **Alternative method:** | | |
| Use a scalar product to find the projection $CN$ (or $DN$) of $BC$ (or $AD$) on $CD$ | *M1 | |
| Obtain $CN = 3$ (or $DN = 3$) | A1 | |
| Use Pythagoras to obtain $BN$ (or $AN$) | DM1 | |
| Obtain answer 3 | A1 | |
| Carry out a correct method to find the area of $ABCD$ and obtain the answer 18 | A1 | |
| **Total** | **5** | |9 The quadrilateral $A B C D$ is a trapezium in which $A B$ and $D C$ are parallel. With respect to the origin $O$, the position vectors of $A , B$ and $C$ are given by $\overrightarrow { O A } = - \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } , \overrightarrow { O B } = \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\overrightarrow { O C } = 2 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }$.
\begin{enumerate}[label=(\alph*)]
\item Given that $\overrightarrow { D C } = 3 \overrightarrow { A B }$, find the position vector of $D$.
\item State a vector equation for the line through $A$ and $B$.
\item Find the distance between the parallel sides and hence find the area of the trapezium.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2021 Q9 [9]}}