| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2023 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Find foot of perpendicular from point to line |
| Difficulty | Standard +0.3 This is a standard vectors question requiring routine techniques: (a) showing lines don't intersect by checking if direction vectors are parallel or finding if a consistent parameter exists, and (b) finding foot of perpendicular using dot product condition. Both parts follow textbook methods with straightforward algebra, making it slightly easier than average for A-level Further Maths. |
| Spec | 1.10b Vectors in 3D: i,j,k notation4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Carry out correct method for finding vector equation for \(AB\) | M1 | |
| Obtain \([\mathbf{r} =]\, \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} + 3\mathbf{k})\) | A1 | OE e.g. \(\mathbf{r} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i}+3\mathbf{j}-3\mathbf{k})\) |
| Equate two pairs of components of general points on their \(AB\) and \(l\) and evaluate \(\lambda\) or \(\mu\) | M1 | \(\begin{pmatrix}1+\lambda\\2-3\lambda\\-2+3\lambda\end{pmatrix} = \begin{pmatrix}1+2\mu\\-1-3\mu\\3+4\mu\end{pmatrix}\) |
| Obtain correct answer for \(\lambda\) or \(\mu\), e.g. \(\lambda = -1\), \(\mu = -2\) | A1 | Correct value from two correct component equations |
| Verify that all three equations are not satisfied and lines fail to intersect (\(\neq\) is sufficient justification e.g. \(0 \neq -3\)) | A1 | Conclusion needs to follow correct values. Hybrid versions possible using j and k to get one parameter then i for the other, or solving two pairs of simultaneous equations and showing results are not the same |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Find \(\overrightarrow{AP}\) for general point \(P\) on \(l\), e.g. \(-3\mathbf{j} + 5\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})\) | B1 | Or equivalent e.g. \(\overrightarrow{PA} = -2\mu\mathbf{i} + (3\mu+3)\mathbf{j} - (4\mu+5)\mathbf{k}\) |
| Calculate scalar product of their \(\overrightarrow{AP}\) and direction vector of \(l\) and equate to zero | M1 | e.g. \(4\mu + (9+9\mu) + (20+16\mu) = 0\). M0 if using \(\overrightarrow{OP}\). M0 if using parallel line through \(A\) |
| Obtain \(\mu = -1\) | A1 | |
| Obtain answer \(-\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) | A1 | Accept coordinates in place of position vector |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Mark | Guidance |
| Find \(\overrightarrow{AP}\) for general point \(P\) on \(l\), e.g. \(-3\mathbf{j} + 5\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})\) | B1 | Or equivalent e.g. \(\overrightarrow{PA} = -2\mu\mathbf{i} + (3\mu+3)\mathbf{j} - (4\mu+5)\mathbf{k}\) |
| Use Pythagoras and differentiate w.r.t. \(\mu\) to obtain value of \(\mu\) corresponding to minimum distance (no need to prove minimum) | M1 | \(\frac{d}{d\mu}\left(4\mu^2 + 9(\mu+1)^2 + (4\mu+5)^2\right) = 0\) |
| Obtain \(\mu = -1\) | A1 | |
| Obtain answer \(-\mathbf{i} + 2\mathbf{j} - \mathbf{k}\) | A1 | Accept coordinates in place of position vector |
## Question 11(a):
| Answer | Mark | Guidance |
|--------|------|----------|
| Carry out correct method for finding vector equation for $AB$ | M1 | |
| Obtain $[\mathbf{r} =]\, \mathbf{i} + 2\mathbf{j} - 2\mathbf{k} + \lambda(\mathbf{i} - 3\mathbf{j} + 3\mathbf{k})$ | A1 | OE e.g. $\mathbf{r} = 2\mathbf{i} - \mathbf{j} + \mathbf{k} + \lambda(-\mathbf{i}+3\mathbf{j}-3\mathbf{k})$ |
| Equate two pairs of components of general points on their $AB$ and $l$ and evaluate $\lambda$ or $\mu$ | M1 | $\begin{pmatrix}1+\lambda\\2-3\lambda\\-2+3\lambda\end{pmatrix} = \begin{pmatrix}1+2\mu\\-1-3\mu\\3+4\mu\end{pmatrix}$ |
| Obtain correct answer for $\lambda$ or $\mu$, e.g. $\lambda = -1$, $\mu = -2$ | A1 | Correct value from two correct component equations |
| Verify that all three equations are not satisfied and lines fail to intersect ($\neq$ is sufficient justification e.g. $0 \neq -3$) | A1 | Conclusion needs to follow correct values. Hybrid versions possible using **j** and **k** to get one parameter then **i** for the other, or solving two pairs of simultaneous equations and showing results are not the same |
---
## Question 11(b):
| Answer | Mark | Guidance |
|--------|------|----------|
| Find $\overrightarrow{AP}$ for general point $P$ on $l$, e.g. $-3\mathbf{j} + 5\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})$ | B1 | Or equivalent e.g. $\overrightarrow{PA} = -2\mu\mathbf{i} + (3\mu+3)\mathbf{j} - (4\mu+5)\mathbf{k}$ |
| Calculate scalar product of their $\overrightarrow{AP}$ and direction vector of $l$ and equate to zero | M1 | e.g. $4\mu + (9+9\mu) + (20+16\mu) = 0$. M0 if using $\overrightarrow{OP}$. M0 if using parallel line through $A$ |
| Obtain $\mu = -1$ | A1 | |
| Obtain answer $-\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ | A1 | Accept coordinates in place of position vector |
**Alternative Method for 11(b):**
| Answer | Mark | Guidance |
|--------|------|----------|
| Find $\overrightarrow{AP}$ for general point $P$ on $l$, e.g. $-3\mathbf{j} + 5\mathbf{k} + \mu(2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k})$ | B1 | Or equivalent e.g. $\overrightarrow{PA} = -2\mu\mathbf{i} + (3\mu+3)\mathbf{j} - (4\mu+5)\mathbf{k}$ |
| Use Pythagoras and differentiate w.r.t. $\mu$ to obtain value of $\mu$ corresponding to minimum distance (no need to prove minimum) | M1 | $\frac{d}{d\mu}\left(4\mu^2 + 9(\mu+1)^2 + (4\mu+5)^2\right) = 0$ |
| Obtain $\mu = -1$ | A1 | |
| Obtain answer $-\mathbf{i} + 2\mathbf{j} - \mathbf{k}$ | A1 | Accept coordinates in place of position vector |11 The points $A$ and $B$ have position vectors $\mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }$ and $2 \mathbf { i } - \mathbf { j } + \mathbf { k }$ respectively. The line $l$ has equation $\mathbf { r } = \mathbf { i } - \mathbf { j } + 3 \mathbf { k } + \mu ( 2 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } )$.
\begin{enumerate}[label=(\alph*)]
\item Show that $l$ does not intersect the line passing through $A$ and $B$.
\item Find the position vector of the foot of the perpendicular from $A$ to $l$.\\
If you use the following lined page to complete the answer(s) to any question(s), the question number(s) must be clearly shown.
\end{enumerate}
\hfill \mbox{\textit{CAIE P3 2023 Q11 [9]}}