| Exam Board | Edexcel |
|---|---|
| Module | PMT Mocks (PMT Mocks) |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Collinearity and ratio division |
| Difficulty | Standard +0.3 Part (a) is a standard collinearity problem requiring students to use the direction vector PQ and apply it to find m—straightforward vector arithmetic. Part (b) involves finding point T using parallel vector conditions and computing magnitude, requiring more steps but still following routine procedures. This is slightly easier than average as it's methodical application of standard vector techniques without requiring novel insight. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10f Distance between points: using position vectors |
| Answer | Marks |
|---|---|
| M1 | Attempts two of the three relevant vectors by subtracting either way round. Example: \(\pm\overrightarrow{PQ} = \pm(i + 2j + 3k)\) or \(\pm\overrightarrow{PR} = \pm(3i + 6j + (m - 2)k)\) or \(\pm\overrightarrow{QR} = \pm(2i + 4j + (m - 5)k)\) |
| dM1 | For the key step in using the fact that if the vectors are parallel, they will be multiples of each other to find m. Example: \(\overrightarrow{QR} = 2\overrightarrow{PQ}\) so \(m - 5 = 2 \times 3\); $\overrightarrow{PR} = 3\overrightar |
(6 marks)
**M1** | Attempts two of the three relevant vectors by subtracting either way round. Example: $\pm\overrightarrow{PQ} = \pm(i + 2j + 3k)$ or $\pm\overrightarrow{PR} = \pm(3i + 6j + (m - 2)k)$ or $\pm\overrightarrow{QR} = \pm(2i + 4j + (m - 5)k)$
**dM1** | For the key step in using the fact that if the vectors are parallel, they will be multiples of each other to find m. Example: $\overrightarrow{QR} = 2\overrightarrow{PQ}$ so $m - 5 = 2 \times 3$; $\overrightarrow{PR} = 3\overrightar13. Relative to a fixed origin $O$
\begin{itemize}
\item the point $P$ has position vector $( 0 , - 1,2 )$
\item the point $Q$ has position vector $( 1,1,5 )$
\item the point $R$ has position vector ( $3,5 , m$ )\\
where $m$ is a constant.\\
Given that $P , Q$ and $R$ lie on a straight line,\\
a. find the value of $m$
\end{itemize}
The line segment $O Q$ is extended to a point $T$ so that $\overrightarrow { R T }$ is parallel to $\overrightarrow { O P }$\\
b. Show that $| \overrightarrow { O T } | = 9 \sqrt { 3 }$.\\
\hfill \mbox{\textit{Edexcel PMT Mocks Q13 [6]}}