| Exam Board | Edexcel |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2021 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Acute angle between line and plane |
| Difficulty | Standard +0.3 This is a straightforward FP1 vectors question with routine steps: (a) reading normal from plane equation, (b) computing a standard cross product, (c) finding a vector in a plane using the given cross product result. While it's Further Maths content, the techniques are mechanical and well-practiced, making it slightly easier than an average A-level question overall. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation4.04c Scalar product: calculate and use for angles4.04g Vector product: a x b perpendicular vector |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\mathbf{n}=\begin{pmatrix}1\\-2\\25\end{pmatrix}\) or any non-zero scalar multiple | B1 | Correct normal vector |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \(\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\1&-2&25\\3&-2&-1\end{vmatrix}=\begin{pmatrix}(-2)(-1)-(-2)(25)\\-((1)(-1)-(3)(25))\\(1)(-2)-(3)(-2)\end{pmatrix}=\ldots\) | M1 | Uses n and \(\mathbf{v}_A\) in cross product; allow slips in coordinates |
| \(=\begin{pmatrix}52\\76\\4\end{pmatrix}=4\begin{pmatrix}13\\19\\1\end{pmatrix}\) | A1 | Correct work leading to a multiple of required vector |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Landing direction is perpendicular to \(\mathbf{n}\times\mathbf{v}_A\) and n; direction given by \(\begin{pmatrix}13\\19\\1\end{pmatrix}\times\begin{pmatrix}1\\-2\\25\end{pmatrix}=\ldots\) | M1 | Recognises \(\mathbf{v}_A\) must be perpendicular to both vector from (b) and n |
| Uses answer from (b) with normal vector to find vector in required direction | M1 | Allow vectors either way round; alternative: find line of intersection of plane containing n and \(\mathbf{v}_A\) with the field plane |
| \(=\begin{pmatrix}477\\-324\\-45\end{pmatrix}\) or any positive multiple e.g. \(\begin{pmatrix}53\\-36\\-5\end{pmatrix}\) or \(\begin{pmatrix}1908\\-1296\\-180\end{pmatrix}\) | A1 | Direction must be correct (positive multiple); if initially incorrect must be adapted |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Any acceptable reason e.g. paths would not be linear / may have lateral movement / could be affected by cross winds / field might not be flat | B1 | Any correct limitation of the model |
# Question 4:
## Part (a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\mathbf{n}=\begin{pmatrix}1\\-2\\25\end{pmatrix}$ or any non-zero scalar multiple | B1 | Correct normal vector |
## Part (b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $\begin{vmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\1&-2&25\\3&-2&-1\end{vmatrix}=\begin{pmatrix}(-2)(-1)-(-2)(25)\\-((1)(-1)-(3)(25))\\(1)(-2)-(3)(-2)\end{pmatrix}=\ldots$ | M1 | Uses **n** and $\mathbf{v}_A$ in cross product; allow slips in coordinates |
| $=\begin{pmatrix}52\\76\\4\end{pmatrix}=4\begin{pmatrix}13\\19\\1\end{pmatrix}$ | A1 | Correct work leading to a multiple of required vector |
## Part (c):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Landing direction is perpendicular to $\mathbf{n}\times\mathbf{v}_A$ and **n**; direction given by $\begin{pmatrix}13\\19\\1\end{pmatrix}\times\begin{pmatrix}1\\-2\\25\end{pmatrix}=\ldots$ | M1 | Recognises $\mathbf{v}_A$ must be perpendicular to both vector from (b) and **n** |
| Uses answer from (b) with normal vector to find vector in required direction | M1 | Allow vectors either way round; alternative: find line of intersection of plane containing **n** and $\mathbf{v}_A$ with the field plane |
| $=\begin{pmatrix}477\\-324\\-45\end{pmatrix}$ or any positive multiple e.g. $\begin{pmatrix}53\\-36\\-5\end{pmatrix}$ or $\begin{pmatrix}1908\\-1296\\-180\end{pmatrix}$ | A1 | Direction must be correct (positive multiple); if initially incorrect must be adapted |
## Part (d):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Any acceptable reason e.g. paths would not be linear / may have lateral movement / could be affected by cross winds / field might not be flat | B1 | Any correct limitation of the model |4.
\begin{figure}[h]
\begin{center}
\includegraphics[alt={},max width=\textwidth]{55803551-f13d-419f-8b51-31642bd20b6a-12_474_1063_264_502}
\captionsetup{labelformat=empty}
\caption{Figure 2}
\end{center}
\end{figure}
A small aircraft is landing in a field.\\
In a model for the landing the aircraft travels in different straight lines before and after it lands, as shown in Figure 2.
The vector $\mathbf { v } _ { \mathbf { A } }$ is in the direction of travel of the aircraft as it approaches the field.\\
The vector $\mathbf { V } _ { \mathbf { L } }$ is in the direction of travel of the aircraft after it lands.\\
With respect to a fixed origin, the field is modelled as the plane with equation
$$x - 2 y + 25 z = 0$$
and
$$\mathbf { v } _ { \mathbf { A } } = \left( \begin{array} { r }
3 \\
- 2 \\
- 1
\end{array} \right)$$
\begin{enumerate}[label=(\alph*)]
\item Write down a vector $\mathbf { n }$ that is a normal vector to the field.
\item Show that $\mathbf { n } \times \mathbf { v } _ { \mathbf { A } } = \lambda \left( \begin{array} { r } 13 \\ 19 \\ 1 \end{array} \right)$, where $\lambda$ is a constant to be determined.
When the aircraft lands it remains in contact with the field and travels in the direction $\mathbf { v } _ { \mathbf { L } }$ The vector $\mathbf { v } _ { \mathbf { L } }$ is in the same plane as both $\mathbf { v } _ { \mathbf { A } }$ and $\mathbf { n }$ as shown in Figure 2.
\item Determine a vector which has the same direction as $\mathbf { V } _ { \mathbf { L } }$
\item State a limitation of the model.
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP1 2021 Q4 [7]}}