| Exam Board | Edexcel |
|---|---|
| Module | M4 (Mechanics 4) |
| Year | 2002 |
| Session | June |
| Marks | 14 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Kinematics with position vectors |
| Difficulty | Challenging +1.2 This M4 mechanics question involves 3D vector kinematics with multiple particles. Part (a) is standard collision detection requiring equating position vectors and solving for time. Part (b) requires understanding relative velocity and setting up simultaneous equations with the parallel vector conditions, which is more sophisticated but follows established M4 techniques. The multi-part structure and need to work with relative velocities elevates it slightly above average difficulty, but it remains a recognizable exam pattern without requiring novel geometric insight. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10e Position vectors: and displacement1.10h Vectors in kinematics: uniform acceleration in vector form |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(\mathbf{r}_P = \begin{pmatrix}1\\-2\\3\end{pmatrix}+t\begin{pmatrix}1\\2\\-1\end{pmatrix},\quad \mathbf{r}_Q = \begin{pmatrix}-1\\2\\-1\end{pmatrix}+t\begin{pmatrix}2\\0\\1\end{pmatrix}\) | B1 | Either position vector |
| Setting equal, solve one component \(\Rightarrow t=2\) | M1 A1 | |
| Showing true for all components \(\Rightarrow\) collide | M1 | |
| \(\mathbf{r} = 3\mathbf{i}+2\mathbf{j}+\mathbf{k}\) | A1 | (5) |
| Answer | Marks | Guidance |
|---|---|---|
| Working/Answer | Marks | Notes |
| \(\mathbf{v}_R - \mathbf{v}_P = \lambda\begin{pmatrix}-5\\4\\-1\end{pmatrix},\quad \mathbf{v}_R-\mathbf{v}_Q = \mu\begin{pmatrix}-2\\2\\-1\end{pmatrix}\) | B1 | |
| \(\mathbf{v}_Q - \mathbf{v}_P = \begin{pmatrix}1\\-2\\2\end{pmatrix} = \lambda\begin{pmatrix}-5\\4\\-1\end{pmatrix}-\mu\begin{pmatrix}-2\\2\\-1\end{pmatrix}\) | M1 | |
| \(-5\lambda+2\mu=1,\quad 4\lambda-2\mu=-2,\quad -\lambda+\mu=2\) | A1 | |
| Solve for \(\lambda=1\) or \(\mu=3\) | M1 A1 | |
| \(\mathbf{v}_R = -4\mathbf{i}+6\mathbf{j}-2\mathbf{k}\) | A1 | |
| \(\begin{pmatrix}3\\2\\1\end{pmatrix}=\begin{pmatrix}a\\b\\c\end{pmatrix}+2\begin{pmatrix}-4\\6\\-2\end{pmatrix}\) | M1 A1 ft | |
| \(\Rightarrow a=11,\,b=-10,\,c=5\) | ||
| At \(t=0\), \(R\) is at \(11\mathbf{i}-10\mathbf{j}+5\mathbf{k}\) | A1 | (9) |
# Question 5:
## Part (a):
| Working/Answer | Marks | Notes |
|---|---|---|
| $\mathbf{r}_P = \begin{pmatrix}1\\-2\\3\end{pmatrix}+t\begin{pmatrix}1\\2\\-1\end{pmatrix},\quad \mathbf{r}_Q = \begin{pmatrix}-1\\2\\-1\end{pmatrix}+t\begin{pmatrix}2\\0\\1\end{pmatrix}$ | B1 | Either position vector |
| Setting equal, solve one component $\Rightarrow t=2$ | M1 A1 | |
| Showing true for all components $\Rightarrow$ collide | M1 | |
| $\mathbf{r} = 3\mathbf{i}+2\mathbf{j}+\mathbf{k}$ | A1 | (5) |
## Part (b):
| Working/Answer | Marks | Notes |
|---|---|---|
| $\mathbf{v}_R - \mathbf{v}_P = \lambda\begin{pmatrix}-5\\4\\-1\end{pmatrix},\quad \mathbf{v}_R-\mathbf{v}_Q = \mu\begin{pmatrix}-2\\2\\-1\end{pmatrix}$ | B1 | |
| $\mathbf{v}_Q - \mathbf{v}_P = \begin{pmatrix}1\\-2\\2\end{pmatrix} = \lambda\begin{pmatrix}-5\\4\\-1\end{pmatrix}-\mu\begin{pmatrix}-2\\2\\-1\end{pmatrix}$ | M1 | |
| $-5\lambda+2\mu=1,\quad 4\lambda-2\mu=-2,\quad -\lambda+\mu=2$ | A1 | |
| Solve for $\lambda=1$ or $\mu=3$ | M1 A1 | |
| $\mathbf{v}_R = -4\mathbf{i}+6\mathbf{j}-2\mathbf{k}$ | A1 | |
| $\begin{pmatrix}3\\2\\1\end{pmatrix}=\begin{pmatrix}a\\b\\c\end{pmatrix}+2\begin{pmatrix}-4\\6\\-2\end{pmatrix}$ | M1 A1 ft | |
| $\Rightarrow a=11,\,b=-10,\,c=5$ | | |
| At $t=0$, $R$ is at $11\mathbf{i}-10\mathbf{j}+5\mathbf{k}$ | A1 | (9) |
---5. At time $t = 0$ particles $P$ and $Q$ start simultaneously from points which have position vectors $( \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } ) \mathrm { m }$ and $( - \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m }$ respectively, relative to a fixed origin $O$. The velocities of $P$ and $Q$ are $( \mathbf { i } + 2 \mathbf { j } - \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ and $( 2 \mathbf { i } + \mathbf { k } ) \mathrm { m } \mathrm { s } ^ { - 1 }$ respectively.
\begin{enumerate}[label=(\alph*)]
\item Show that $P$ and $Q$ collide and find the position vector of the point at which they collide.
A third particle $R$ moves in such a way that its velocity relative to $P$ is parallel to the vector ( $- 5 \mathbf { i } + 4 \mathbf { j } - \mathbf { k }$ ) and its velocity relative to $Q$ is parallel to the vector $( - 2 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } )$.
Given that all three particles collide simultaneously, find
\item \begin{enumerate}[label=(\roman*)]
\item the velocity of $R$,
\item the position vector of $R$ at time $t = 0$.
\end{enumerate}\end{enumerate}
\hfill \mbox{\textit{Edexcel M4 2002 Q5 [14]}}