| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2002 |
| Session | November |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Perpendicular distance point to line |
| Difficulty | Standard +0.3 This is a standard multi-part vectors question requiring routine techniques: finding angle between lines using dot product, proving intersection by solving parametric equations, and calculating perpendicular distance using the vector formula. All methods are textbook exercises with straightforward arithmetic, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10f Distance between points: using position vectors1.10g Problem solving with vectors: in geometry |
| Answer | Marks | Guidance |
|---|---|---|
| Content | Mark | Guidance |
| (i) Find a direction vector for \(AB\) or \(CD\) e.g. \(\overrightarrow{AB} = \mathbf{i} - 2\mathbf{j} - 3\mathbf{k}\) or \(\overrightarrow{CD} = -2\mathbf{i} - \mathbf{j} - 4\mathbf{k}\) | B1 | |
| EITHER: Carry out the correct process for evaluating the scalar product of two relevant vectors in component form | M1 | |
| Evaluate \(\cos^{-1}\left(\frac{\overrightarrow{AB}\overrightarrow{CD}}{ | \overrightarrow{AB} | |
| Obtain final answer \(45.6°\), or \(0.796\) radians, correctly | A1 | |
| OR: Calculate the sides of a relevant triangle using the correct method | M1 | |
| Use the cosine rule to calculate a relevant angle | M1 | |
| Obtain final answer \(45.6°\), or \(0.796\) radians, correctly | A1 | [SR: if a vector is incorrectly stated with all signs reversed and 45.6° is obtained, award B0M1M1A1.] [SR: if 45.6° is followed by 44.4° as final answer, award A0.] |
| (ii) EITHER: State both line equations e.g. \(4\mathbf{i} + \mathbf{k} + \lambda(\mathbf{2i} - 3\mathbf{k}) = 0\) and \(\mathbf{i} + \mathbf{j} + \mu(\mathbf{2i} + \mathbf{j} + 4\mathbf{k})\) | B1✓ | |
| Equate components and solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain value \(\lambda = -1\) or \(\mu = 1\) | A1 | |
| Verify that all equations are satisfied, so that the lines do intersect, or equivalent | A1 | [SR: if both lines have the same parameter, award B1M1 if the equations are inconsistent and B1M1A1 if the equations are consistent and shown to be so.] |
| OR: State both line equations in Cartesian form | B1✓ | |
| Solve simultaneous equations for a pair of unknowns e.g. \(x\) and \(y\) | M1 | |
| Obtain a correct pair e.g. \(x = 3, y = 2\) | A1 | |
| Obtain the third unknown e.g. \(z = 4\) and verify the lines intersect | A1 | |
| OR: Find one of \(\overrightarrow{CA}, \overrightarrow{CB}, \overrightarrow{DA}, \overrightarrow{DB}\), \(\ldots\) e.g. \(\overrightarrow{CA} = -3\mathbf{i} - \mathbf{j} + \mathbf{k}\) | B1 | |
| Carry out correct process for evaluating a relevant scalar triple product e.g. \(\overrightarrow{CA} \cdot (\overrightarrow{AB} \times \overrightarrow{CD})\) | M1 | |
| Show the value is zero | A1 | |
| State that (a) this result implies the lines are coplanar, (b) the lines are not parallel, and thus the lines intersect (condone omission of one of (a) and (b)) | A1 | |
| OR: Carry out correct method for finding a normal to the plane through three of the points | M1 | |
| Obtain a correct normal vector | A1 | |
| Obtain a correct equation e.g. \(x + 2y - z = 3\) for the plane of \(A, B, C\) | A1 | |
| Verify that the fourth point lies in the plane and conclude that the lines intersect | A1 | |
| OR: State a relevant plane equation e.g. \(\mathbf{r} = 4\mathbf{i} + \mathbf{k} + \lambda(\mathbf{2i} - 3\mathbf{k}) + \mu(-3\mathbf{i} + \mathbf{j} - \mathbf{k})\) for the plane of \(A, B, C\) | B1✓ | |
| Set up equations in \(\lambda\) and \(\mu\) using components of the fourth point, and solve for \(\lambda\) or \(\mu\) | M1 | |
| Obtain value \(\lambda = 1\) or \(\mu = 2\) | A1 | |
| Verify that all equations are satisfied and conclude that the lines intersect | A1 | Max 4 marks |
| (iii) EITHER: Find \(\overrightarrow{PQ}\) for a general point \(Q\) on \(AB\) e.g. \(3\mathbf{i} - 5\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{2i} - 3\mathbf{k})\) | B1✓ | |
| Calculate \(\overrightarrow{PQ} \cdot \overrightarrow{AB}\) correctly and equate to zero | M1 | |
| Solve for \(\lambda\) obtaining \(\lambda = -2\) | A1 | |
| Show correctly that \(\overrightarrow{PQ} = \sqrt{3}\), the given answer | A1 | Max 4 marks |
| OR: State \(\overrightarrow{AP}\) (or \(\overrightarrow{BP}\)) and \(\overrightarrow{AB}\) in component form | B1✓ | |
| Carry out correct method for finding their vector product | M1 | |
| Obtain correct answer e.g. \(\overrightarrow{AP} \times \overrightarrow{AB} = -5\mathbf{i} - 4\mathbf{j} + \mathbf{k}\) | A1 | |
| Divide modulus by \( | \overrightarrow{AB} | \) and obtain the given answer \(\sqrt{3}\) |
| OR: State \(\overrightarrow{AP}\) (or \(\overrightarrow{BP}\)) and \(\overrightarrow{AB}\) in component form | B1✓ | |
| Carry out correct method for finding the projection of \(\overrightarrow{AP}\) (or \(\overrightarrow{BP}\)) on \(AB\) i.e. \(\frac{\overrightarrow{AP} \cdot \overrightarrow{AB}}{ | \overrightarrow{AB} | }\) |
| Obtain correct answer e.g. \(AN = \frac{-28}{\sqrt{14}}\) or \(BN = \frac{-42}{\sqrt{14}}\) | A1 | |
| Show correctly that \(PN = \sqrt{3}\), the given answer | A1 | |
| OR: State two of \(\overrightarrow{AP}, \overrightarrow{BP}, \overrightarrow{AB}\) in component form | B1✓ | |
| Use the cosine rule in triangle \(ABP\), or scalar product, to find the cosine of \(A, B,\) or \(P\) | M1 | |
| Obtain correct answer e.g. \(\cos A = \frac{-28}{\sqrt{14}\sqrt{59}}\) | A1 | |
| Deduce the exact length of the perpendicular from \(P\) to \(AB\) is \(\sqrt{3}\), the given answer | A1 |
| Content | Mark | Guidance |
|---------|------|----------|
| **(i)** Find a direction vector for $AB$ or $CD$ e.g. $\overrightarrow{AB} = \mathbf{i} - 2\mathbf{j} - 3\mathbf{k}$ or $\overrightarrow{CD} = -2\mathbf{i} - \mathbf{j} - 4\mathbf{k}$ | B1 | |
| **EITHER:** Carry out the correct process for evaluating the scalar product of two relevant vectors in component form | M1 | |
| Evaluate $\cos^{-1}\left(\frac{\overrightarrow{AB}\overrightarrow{CD}}{|\overrightarrow{AB}||\overrightarrow{CD}|}\right)$ using the correct method for the moduli | M1 | |
| Obtain final answer $45.6°$, or $0.796$ radians, correctly | A1 | |
| **OR:** Calculate the sides of a relevant triangle using the correct method | M1 | |
| Use the cosine rule to calculate a relevant angle | M1 | |
| Obtain final answer $45.6°$, or $0.796$ radians, correctly | A1 | [SR: if a vector is incorrectly stated with all signs reversed and 45.6° is obtained, award B0M1M1A1.] [SR: if 45.6° is followed by 44.4° as final answer, award A0.] | Max 4 marks |
| | | |
| **(ii)** **EITHER:** State both line equations e.g. $4\mathbf{i} + \mathbf{k} + \lambda(\mathbf{2i} - 3\mathbf{k}) = 0$ and $\mathbf{i} + \mathbf{j} + \mu(\mathbf{2i} + \mathbf{j} + 4\mathbf{k})$ | B1✓ | |
| Equate components and solve for $\lambda$ or for $\mu$ | M1 | |
| Obtain value $\lambda = -1$ or $\mu = 1$ | A1 | |
| Verify that all equations are satisfied, so that the lines do intersect, or equivalent | A1 | [SR: if both lines have the same parameter, award B1M1 if the equations are inconsistent and B1M1A1 if the equations are consistent and shown to be so.] | |
| | | |
| **OR:** State both line equations in Cartesian form | B1✓ | |
| Solve simultaneous equations for a pair of unknowns e.g. $x$ and $y$ | M1 | |
| Obtain a correct pair e.g. $x = 3, y = 2$ | A1 | |
| Obtain the third unknown e.g. $z = 4$ and verify the lines intersect | A1 | |
| | | |
| **OR:** Find one of $\overrightarrow{CA}, \overrightarrow{CB}, \overrightarrow{DA}, \overrightarrow{DB}$, $\ldots$ e.g. $\overrightarrow{CA} = -3\mathbf{i} - \mathbf{j} + \mathbf{k}$ | B1 | |
| Carry out correct process for evaluating a relevant scalar triple product e.g. $\overrightarrow{CA} \cdot (\overrightarrow{AB} \times \overrightarrow{CD})$ | M1 | |
| Show the value is zero | A1 | |
| State that (a) this result implies the lines are coplanar, (b) the lines are not parallel, and thus the lines intersect (condone omission of one of (a) and (b)) | A1 | |
| | | |
| **OR:** Carry out correct method for finding a normal to the plane through three of the points | M1 | |
| Obtain a correct normal vector | A1 | |
| Obtain a correct equation e.g. $x + 2y - z = 3$ for the plane of $A, B, C$ | A1 | |
| Verify that the fourth point lies in the plane and conclude that the lines intersect | A1 | |
| | | |
| **OR:** State a relevant plane equation e.g. $\mathbf{r} = 4\mathbf{i} + \mathbf{k} + \lambda(\mathbf{2i} - 3\mathbf{k}) + \mu(-3\mathbf{i} + \mathbf{j} - \mathbf{k})$ for the plane of $A, B, C$ | B1✓ | |
| Set up equations in $\lambda$ and $\mu$ using components of the fourth point, and solve for $\lambda$ or $\mu$ | M1 | |
| Obtain value $\lambda = 1$ or $\mu = 2$ | A1 | |
| Verify that all equations are satisfied and conclude that the lines intersect | A1 | Max 4 marks |
| | | |
| **(iii)** **EITHER:** Find $\overrightarrow{PQ}$ for a general point $Q$ on $AB$ e.g. $3\mathbf{i} - 5\mathbf{j} - 5\mathbf{k} + \lambda(\mathbf{2i} - 3\mathbf{k})$ | B1✓ | |
| Calculate $\overrightarrow{PQ} \cdot \overrightarrow{AB}$ correctly and equate to zero | M1 | |
| Solve for $\lambda$ obtaining $\lambda = -2$ | A1 | |
| Show correctly that $\overrightarrow{PQ} = \sqrt{3}$, the given answer | A1 | Max 4 marks |
| | | |
| **OR:** State $\overrightarrow{AP}$ (or $\overrightarrow{BP}$) and $\overrightarrow{AB}$ in component form | B1✓ | |
| Carry out correct method for finding their vector product | M1 | |
| Obtain correct answer e.g. $\overrightarrow{AP} \times \overrightarrow{AB} = -5\mathbf{i} - 4\mathbf{j} + \mathbf{k}$ | A1 | |
| Divide modulus by $|\overrightarrow{AB}|$ and obtain the given answer $\sqrt{3}$ | A1 | |
| | | |
| **OR:** State $\overrightarrow{AP}$ (or $\overrightarrow{BP}$) and $\overrightarrow{AB}$ in component form | B1✓ | |
| Carry out correct method for finding the projection of $\overrightarrow{AP}$ (or $\overrightarrow{BP}$) on $AB$ i.e. $\frac{\overrightarrow{AP} \cdot \overrightarrow{AB}}{|\overrightarrow{AB}|}$ | M1 | |
| Obtain correct answer e.g. $AN = \frac{-28}{\sqrt{14}}$ or $BN = \frac{-42}{\sqrt{14}}$ | A1 | |
| Show correctly that $PN = \sqrt{3}$, the given answer | A1 | |
| | | |
| **OR:** State two of $\overrightarrow{AP}, \overrightarrow{BP}, \overrightarrow{AB}$ in component form | B1✓ | |
| Use the cosine rule in triangle $ABP$, or scalar product, to find the cosine of $A, B,$ or $P$ | M1 | |
| Obtain correct answer e.g. $\cos A = \frac{-28}{\sqrt{14}\sqrt{59}}$ | A1 | |
| Deduce the exact length of the perpendicular from $P$ to $AB$ is $\sqrt{3}$, the given answer | A1 | |10 With respect to the origin $O$, the points $A , B , C , D$ have position vectors given by
$$\overrightarrow { O A } = 4 \mathbf { i } + \mathbf { k } , \quad \overrightarrow { O B } = 5 \mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , \quad \overrightarrow { O C } = \mathbf { i } + \mathbf { j } , \quad \overrightarrow { O D } = - \mathbf { i } - 4 \mathbf { k }$$
(i) Calculate the acute angle between the lines $A B$ and $C D$.\\
(ii) Prove that the lines $A B$ and $C D$ intersect.\\
(iii) The point $P$ has position vector $\mathbf { i } + 5 \mathbf { j } + 6 \mathbf { k }$. Show that the perpendicular distance from $P$ to the line $A B$ is equal to $\sqrt { } 3$.
\hfill \mbox{\textit{CAIE P3 2002 Q10 [12]}}