| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2004 |
| Session | November |
| Marks | 10 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Cross Product & Distances |
| Type | Common perpendicular to two skew lines |
| Difficulty | Challenging +1.3 This is a structured multi-part vectors question requiring standard techniques: showing lines are skew (equating components and checking for inconsistency), finding a point on a line using perpendicularity conditions (dot product = 0), and verification. While it involves several steps and the cross product/perpendicular concepts, each part follows predictable methods taught in Further Maths. The 'common perpendicular' context adds slight conceptual depth beyond routine exercises, but the execution is methodical rather than requiring novel insight. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| (i) EITHER: Express general point of \(l\) or \(m\) in component form e.g. \((2+s, -1+s, 4-s)\) or \((-2-2t, 2+t, 1+t)\) | B1 | |
| Equate at least two pairs of components and solve for \(s\) or for \(t\) | M1 | |
| Obtain correct answer for \(s\) or \(t\) (possible answers are \(\frac{2}{3}, 10,\) or \(3\) for \(s\) and \(-\frac{7}{3}, -7,\) or \(0\) for \(t\)) | A1 | |
| Verify that all three component equations are not satisfied | A1 | |
| OR: State a Cartesian equation for \(l\) or for \(m\), e.g. \(\frac{x-2}{1} = \frac{y-(-1)}{1} = \frac{z-4}{-1}\) for \(l\) | B1 | |
| Solve a pair of equations for a pair of values, e.g. \(x\) and \(y\) | M1 | |
| Obtain a pair of correct answers, e.g. \(x = -\frac{8}{3}\) and \(y = -\frac{1}{3}\) | A1 | |
| Find corresponding remaining values, e.g. of \(z\), and show lines do not intersect | A1 | |
| OR: Form a relevant triple scalar product, e.g. \((4i-3j+3k) \cdot ((i+j-k) \times (-2i+j+k))\) | B1 | |
| Attempt to use correct method of evaluation | M1 | |
| Obtain at least two correct simplified terms of the three terms of the complete expansion of the triple product or of the corresponding determinant | A1 | |
| Obtain correct non-zero value, e.g. 14, and state that the lines cannot intersect | A1 | Total: 4 marks |
| (ii) EITHER: Express \(\overrightarrow{PQ}\) or \(\overrightarrow{QP}\) in terms of \(s\) in any correct form e.g. \(-si+(1-s)j+(-5-s)k\) | B1 | |
| Equate its scalar product with a direction vector for \(l\) to zero, obtaining a linear equation in \(s\) | M1 | |
| Solve for \(s\) | M1 | |
| Obtain \(s = 2\) and \(\overrightarrow{OP}\) is \(4i+j+2k\) | A1 | |
| OR: Take a point A on \(l\), e.g. \((2, -1, 4)\), and use scalar product to calculate \(AP\), the length of the projection of \(AQ\) onto \(l\) | M1 | |
| Obtain answer \(AP = 2\sqrt{3}\), or equivalent | A1 | |
| Carry out method for finding \(\overrightarrow{OP}\) | M1 | |
| Obtain answer \(4i+j+2k\) | A1 | Total: 4 marks |
| (iii) Show that \(Q\) is the point on \(m\) with parameter \(t = –2\), or that \((2, 0, –1)\) satisfies the Cartesian equation of \(m\) | B1 | |
| Show that \(PQ\) is perpendicular to \(m\) e.g. by verifying fully that \((-2i-j-3k) \cdot (-2i+j+k) = 0\) | B1 | Total: 2 marks |
**(i)** **EITHER:** Express general point of $l$ or $m$ in component form e.g. $(2+s, -1+s, 4-s)$ or $(-2-2t, 2+t, 1+t)$ | B1 | |
Equate at least two pairs of components and solve for $s$ or for $t$ | M1 | |
Obtain correct answer for $s$ or $t$ (possible answers are $\frac{2}{3}, 10,$ or $3$ for $s$ and $-\frac{7}{3}, -7,$ or $0$ for $t$) | A1 | |
Verify that all three component equations are not satisfied | A1 | |
**OR:** State a Cartesian equation for $l$ or for $m$, e.g. $\frac{x-2}{1} = \frac{y-(-1)}{1} = \frac{z-4}{-1}$ for $l$ | B1 | |
Solve a pair of equations for a pair of values, e.g. $x$ and $y$ | M1 | |
Obtain a pair of correct answers, e.g. $x = -\frac{8}{3}$ and $y = -\frac{1}{3}$ | A1 | |
Find corresponding remaining values, e.g. of $z$, and show lines do not intersect | A1 | |
**OR:** Form a relevant triple scalar product, e.g. $(4i-3j+3k) \cdot ((i+j-k) \times (-2i+j+k))$ | B1 | |
Attempt to use correct method of evaluation | M1 | |
Obtain at least two correct simplified terms of the three terms of the complete expansion of the triple product or of the corresponding determinant | A1 | |
Obtain correct non-zero value, e.g. 14, and state that the lines cannot intersect | A1 | **Total: 4 marks** |
**(ii)** **EITHER:** Express $\overrightarrow{PQ}$ or $\overrightarrow{QP}$ in terms of $s$ in any correct form e.g. $-si+(1-s)j+(-5-s)k$ | B1 | |
Equate its scalar product with a direction vector for $l$ to zero, obtaining a linear equation in $s$ | M1 | |
Solve for $s$ | M1 | |
Obtain $s = 2$ and $\overrightarrow{OP}$ is $4i+j+2k$ | A1 | |
**OR:** Take a point A on $l$, e.g. $(2, -1, 4)$, and use scalar product to calculate $AP$, the length of the projection of $AQ$ onto $l$ | M1 | |
Obtain answer $AP = 2\sqrt{3}$, or equivalent | A1 | |
Carry out method for finding $\overrightarrow{OP}$ | M1 | |
Obtain answer $4i+j+2k$ | A1 | **Total: 4 marks** |
**(iii)** Show that $Q$ is the point on $m$ with parameter $t = –2$, or that $(2, 0, –1)$ satisfies the Cartesian equation of $m$ | B1 | |
Show that $PQ$ is perpendicular to $m$ e.g. by verifying fully that $(-2i-j-3k) \cdot (-2i+j+k) = 0$ | B1 | **Total: 2 marks** |9 The lines $l$ and $m$ have vector equations
$$\mathbf { r } = 2 \mathbf { i } - \mathbf { j } + 4 \mathbf { k } + s ( \mathbf { i } + \mathbf { j } - \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = - 2 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } + t ( - 2 \mathbf { i } + \mathbf { j } + \mathbf { k } )$$
respectively.\\
(i) Show that $l$ and $m$ do not intersect.
The point $P$ lies on $l$ and the point $Q$ has position vector $2 \mathbf { i } - \mathbf { k }$.\\
(ii) Given that the line $P Q$ is perpendicular to $l$, find the position vector of $P$.\\
(iii) Verify that $Q$ lies on $m$ and that $P Q$ is perpendicular to $m$.
\hfill \mbox{\textit{CAIE P3 2004 Q9 [10]}}