| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2016 |
| Session | June |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Show lines are skew (non-intersecting) |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring routine techniques: finding the line through two points, checking for intersection by equating components (which leads to inconsistent equations), and using the perpendicular distance formula. While it involves multiple steps, each technique is well-practiced and follows textbook methods without requiring novel insight or complex problem-solving. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting4.04h Shortest distances: between parallel lines and between skew lines |
| Answer | Marks | Guidance |
|---|---|---|
| (i) State a correct equation for \(\overrightarrow{AB}\) in any form, e.g. \(r = \vec{i} + \vec{j} + \vec{k} + \lambda(\vec{i} - \vec{j} + 2\vec{k})\), or equivalent | B1 | |
| Equate at least two pairs of components of \(\overrightarrow{AB}\) and \(l\) and solve for \(\lambda\) or for \(\mu\) | M1 | |
| Obtain correct answer for \(\lambda\) or for \(\mu\), e.g. \(\lambda = -1\) or \(\mu = 2\) | A1 | |
| Show that not all three equations are not satisfied and that the lines do not intersect | A1 | [4] |
| (ii) EITHER: Find \(\overrightarrow{AP}\) (or \(\overrightarrow{PA}\)) for a general point \(P\) on \(l\), e.g. \((1-\mu)\vec{i} + (-3+2\mu)\vec{j} + (-2+\mu)\vec{k}\) | B1 | |
| Calculate the scalar product of \(\overrightarrow{AP}\) and a direction vector for \(l\) and equate to zero | M1 | |
| Solve and obtain \(\mu = \frac{3}{2}\) | A1 | |
| Carry out a method to calculate \(AP\) when \(\mu = \frac{3}{2}\) | M1 | |
| Obtain the given answer \(\frac{1}{\sqrt{2}}\) correctly | A1 | |
| OR 1: Find \(\overrightarrow{AP}\) (or \(\overrightarrow{PA}\)) for a general point \(P\) on \(l\) | B1 | |
| Use correct method to express \(AP^2\) (or \(AP\)) in terms of \(\mu\) | M1 | |
| Obtain a correct expression in any form, e.g. \((1-\mu)^2 + (-3+2\mu)^2 + (-2+\mu)^2\) | A1 | |
| Carry out a complete method for finding its minimum | M1 | |
| Obtain the given answer correctly | A1 | |
| OR 2: Calling \((2, -2, -1)\) \(C\), state \(\overrightarrow{AC}\) (or \(\overrightarrow{CA}\)) in component form, e.g. \(\vec{i} - 3\vec{j} - 2\vec{k}\) | B1 | |
| Use a scalar product to find the projection of \(\overrightarrow{AC}\) (or \(\overrightarrow{CA}\)) on \(l\) | M1 | |
| Obtain correct answer in any form, e.g. \(-\frac{7}{\sqrt{6}}\) | A1 | |
| Use Pythagoras to find the perpendicular | M1 | |
| Obtain the given answer correctly | A1 | |
| OR 3: State \(\overrightarrow{AC}\) (or \(\overrightarrow{CA}\)) in component form | B1 | |
| Calculate vector product of \(\overrightarrow{AC}\) and a direction vector for \(l\), e.g. \((\vec{i} - 3\vec{j} - 2\vec{k}) \times (\vec{i} + 2\vec{j} + \vec{k})\) | M1 | |
| Obtain correct answer in any form, e.g. \(\vec{i} + \vec{j} - \vec{k}\) | A1 | |
| Divide modulus of the product by that of the direction vector | M1 | |
| Obtain the given answer correctly | A1 | [5] |
**(i)** State a correct equation for $\overrightarrow{AB}$ in any form, e.g. $r = \vec{i} + \vec{j} + \vec{k} + \lambda(\vec{i} - \vec{j} + 2\vec{k})$, or equivalent | B1 |
Equate at least two pairs of components of $\overrightarrow{AB}$ and $l$ and solve for $\lambda$ or for $\mu$ | M1 |
Obtain correct answer for $\lambda$ or for $\mu$, e.g. $\lambda = -1$ or $\mu = 2$ | A1 |
Show that not all three equations are not satisfied and that the lines do not intersect | A1 | [4]
**(ii)** **EITHER:** Find $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) for a general point $P$ on $l$, e.g. $(1-\mu)\vec{i} + (-3+2\mu)\vec{j} + (-2+\mu)\vec{k}$ | B1 |
Calculate the scalar product of $\overrightarrow{AP}$ and a direction vector for $l$ and equate to zero | M1 |
Solve and obtain $\mu = \frac{3}{2}$ | A1 |
Carry out a method to calculate $AP$ when $\mu = \frac{3}{2}$ | M1 |
Obtain the given answer $\frac{1}{\sqrt{2}}$ correctly | A1 |
**OR 1:** Find $\overrightarrow{AP}$ (or $\overrightarrow{PA}$) for a general point $P$ on $l$ | B1 |
Use correct method to express $AP^2$ (or $AP$) in terms of $\mu$ | M1 |
Obtain a correct expression in any form, e.g. $(1-\mu)^2 + (-3+2\mu)^2 + (-2+\mu)^2$ | A1 |
Carry out a complete method for finding its minimum | M1 |
Obtain the given answer correctly | A1 |
**OR 2:** Calling $(2, -2, -1)$ $C$, state $\overrightarrow{AC}$ (or $\overrightarrow{CA}$) in component form, e.g. $\vec{i} - 3\vec{j} - 2\vec{k}$ | B1 |
Use a scalar product to find the projection of $\overrightarrow{AC}$ (or $\overrightarrow{CA}$) on $l$ | M1 |
Obtain correct answer in any form, e.g. $-\frac{7}{\sqrt{6}}$ | A1 |
Use Pythagoras to find the perpendicular | M1 |
Obtain the given answer correctly | A1 |
**OR 3:** State $\overrightarrow{AC}$ (or $\overrightarrow{CA}$) in component form | B1 |
Calculate vector product of $\overrightarrow{AC}$ and a direction vector for $l$, e.g. $(\vec{i} - 3\vec{j} - 2\vec{k}) \times (\vec{i} + 2\vec{j} + \vec{k})$ | M1 |
Obtain correct answer in any form, e.g. $\vec{i} + \vec{j} - \vec{k}$ | A1 |
Divide modulus of the product by that of the direction vector | M1 |
Obtain the given answer correctly | A1 | [5]
---8 The points $A$ and $B$ have position vectors, relative to the origin $O$, given by $\overrightarrow { O A } = \mathbf { i } + \mathbf { j } + \mathbf { k }$ and $\overrightarrow { O B } = 2 \mathbf { i } + 3 \mathbf { k }$. The line $l$ has vector equation $\mathbf { r } = 2 \mathbf { i } - 2 \mathbf { j } - \mathbf { k } + \mu ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )$.\\
(i) Show that the line passing through $A$ and $B$ does not intersect $l$.\\
(ii) Show that the length of the perpendicular from $A$ to $l$ is $\frac { 1 } { \sqrt { 2 } }$.
\hfill \mbox{\textit{CAIE P3 2016 Q8 [9]}}