| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Parallel and perpendicular lines |
| Difficulty | Moderate -0.8 This is a straightforward vector question requiring basic operations: finding direction vectors by subtraction, checking if they're parallel by scalar multiplication, and identifying a trapezium. All steps are routine with no problem-solving insight needed, making it easier than average but not trivial since it involves multiple 3D vectors and a geometric conclusion. |
| 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 multiplication1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| \(\overline{AB} = -\overrightarrow{OA} + \overrightarrow{OB} = 2\mathbf{i} - 4\mathbf{j} - 7\mathbf{k} - \mathbf{i} + 3\mathbf{j} + 8\mathbf{k} = \mathbf{i} - \mathbf{j} + \mathbf{k}\) | M1 | Attempts to subtract either way round of either \(\overline{AB}\) or \(\overline{CD}\) |
| \(\overline{CD} = -\overrightarrow{OC} + \overrightarrow{OD} = -\mathbf{i} - \mathbf{j} - 4\mathbf{k} - \mathbf{i} + 3\mathbf{j} + 2\mathbf{k} = -2\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}\) | A1 | Correctly obtains either \(\overline{AB}\) or \(\overline{CD}\) |
| A1 | Correctly obtains both \(\overline{AB}\) and \(\overline{CD}\) | |
| \(\overline{AB}\) and \(\overline{CD}\) are parallel as \(\overline{CD} = -2\overline{AB}\), and \(\overline{AB} : \overline{CD} = 1 : 2\) | B1 | States the ratio of \(\overline{AB} : \overline{CD} = 1 : 2\) |
| Answer | Marks | Guidance |
|---|---|---|
| A quadrilateral with one set of parallel sides is a trapezium | B1 | describes that the quadrilateral \(ABCD\) is a trapezium |
**Part a:** Show that $\overline{AB}$ and $\overline{CD}$ are parallel and the ratio $\overline{AB} : \overline{CD}$ in its simplest form
| $\overline{AB} = -\overrightarrow{OA} + \overrightarrow{OB} = 2\mathbf{i} - 4\mathbf{j} - 7\mathbf{k} - \mathbf{i} + 3\mathbf{j} + 8\mathbf{k} = \mathbf{i} - \mathbf{j} + \mathbf{k}$ | M1 | Attempts to subtract either way round of either $\overline{AB}$ or $\overline{CD}$ |
| --- | --- | --- |
| $\overline{CD} = -\overrightarrow{OC} + \overrightarrow{OD} = -\mathbf{i} - \mathbf{j} - 4\mathbf{k} - \mathbf{i} + 3\mathbf{j} + 2\mathbf{k} = -2\mathbf{i} + 2\mathbf{j} - 2\mathbf{k}$ | A1 | Correctly obtains either $\overline{AB}$ or $\overline{CD}$ |
| | A1 | Correctly obtains both $\overline{AB}$ and $\overline{CD}$ |
| $\overline{AB}$ and $\overline{CD}$ are parallel as $\overline{CD} = -2\overline{AB}$, and $\overline{AB} : \overline{CD} = 1 : 2$ | B1 | States the ratio of $\overline{AB} : \overline{CD} = 1 : 2$ |
**Part b:** Hence describe the quadrilateral $ABCD$
| A quadrilateral with one set of parallel sides is a trapezium | B1 | describes that the quadrilateral $ABCD$ is a trapezium |
**(4) marks + (1) mark = (5) marks total for Question 3**
---3. Relative to a fixed origin,
\begin{itemize}
\item point $A$ has position vector $- 2 \mathbf { i } + 4 \mathbf { j } + 7 \mathbf { k }$
\item point $B$ has position vector $- \mathbf { i } + 3 \mathbf { j } + 8 \mathbf { k }$
\item point $C$ has position vector $\mathbf { i } + \mathbf { j } + 4 \mathbf { k }$
\item point $D$ has position vector $- \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k }$\\
a. Show that $\overrightarrow { A B }$ and $\overrightarrow { C D }$ are parallel and the ratio $\overrightarrow { A B } : \overrightarrow { C D }$ in its simplest form.\\
b. Hence describe the quadrilateral $A B C D$.
\end{itemize}
\hfill \mbox{\textit{Edexcel PMT Mocks Q3 [5]}}