| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2003 |
| Session | November |
| Marks | 11 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Line intersection with line |
| Difficulty | Standard +0.3 This is a standard two-part vectors question requiring routine techniques: equating parametric equations to find intersection (solving simultaneous equations), then finding a plane equation using cross product of direction vectors. While it involves multiple steps and vector manipulation, these are well-practiced procedures from the syllabus with no novel insight required, making it slightly easier than average. |
| Spec | 4.04a Line equations: 2D and 3D, cartesian and vector forms4.04b Plane equations: cartesian and vector forms4.04e Line intersections: parallel, skew, or intersecting |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Express general point of \(l\) or \(m\) in component form e.g. \((1+2s, s, -2+3s)\) or \((6+t, -5-2t, 4+t)\) | B1 | |
| Equate at least two corresponding pairs of components and attempt to solve for \(s\) or \(t\) | M1 | |
| Obtain \(s = 1\) or \(t = -3\) | A1 | |
| Verify that all three component equations are satisfied | A1 | |
| Obtain position vector \(3\mathbf{i} + \mathbf{j} + \mathbf{k}\) of intersection point, or equivalent | A1 | The follow through is on \(3\mathbf{i} + \mathbf{j} + \mathbf{k}\) only |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| EITHER: Use scalar product to obtain \(2a + b + 3c = 0\) and \(a - 2b + c = 0\) | B1 | |
| Solve and find one ratio e.g. \(a : b\) | M1 | |
| State one correct ratio | A1 | |
| Obtain answer \(a : b : c = 7 : 1 : -5\), or equivalent | A1 | |
| Substitute coordinates of a relevant point and values of \(a\), \(b\) and \(c\) in general equation of plane and calculate \(d\) | M1 | |
| Obtain answer \(7x + y - 5z = 17\), or equivalent | A1 | |
| OR: Using two points on \(l\) and one on \(m\) (or vice versa) state three simultaneous equations in \(a, b, c\) and \(d\) e.g. \(3a + b + c = d\), \(a - 2c = d\) and \(6a - 5b + 4c = d\) | B1\(\sqrt{}\) | |
| Solve and find one ratio e.g. \(a : b\) | M1 | |
| State one correct ratio | A1 | |
| Obtain a ratio of three unknowns e.g. \(a : b : c = 7 : 1 : -5\), or equivalent | A1 | |
| Use coordinates of a relevant point and found ratio to find fourth unknown e.g. \(d\) | M1 | |
| Obtain answer \(7x + y - 5z = 17\), or equivalent | A1 | |
| OR: Form a correct 2-parameter equation for the plane, e.g. \(\mathbf{r} = \mathbf{i} - 2\mathbf{k} + \lambda(2\mathbf{i}+\mathbf{j}+3\mathbf{k}) + \mu(\mathbf{i}-2\mathbf{j}+\mathbf{k})\) | B1\(\sqrt{}\) | |
| State 3 equations in \(x, y, z, \lambda\) and \(\mu\) | M1 | |
| State 3 correct equations | A1\(\sqrt{}\) | |
| Eliminate \(\lambda\) and \(\mu\) | M1 | |
| Obtain equation in any correct unsimplified form | A1 | |
| Obtain \(7x + y - 5z = 17\), or equivalent | A1 | |
| OR: Attempt to calculate vector product of vectors parallel to \(l\) and \(m\) | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct product, e.g. \(7\mathbf{i}+\mathbf{j}-5\mathbf{z}\) | A1 | |
| State that the plane has equation of the form \(7x + y - 5z = d\) | A1\(\sqrt{}\) | |
| Substitute coordinates of a relevant point and calculate \(d\) | M1 | |
| Obtain answer \(7x + y - 5z = 17\), or equivalent | A1 |
## Question 10(i):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Express general point of $l$ or $m$ in component form e.g. $(1+2s, s, -2+3s)$ or $(6+t, -5-2t, 4+t)$ | B1 | |
| Equate at least two corresponding pairs of components and attempt to solve for $s$ or $t$ | M1 | |
| Obtain $s = 1$ or $t = -3$ | A1 | |
| Verify that all three component equations are satisfied | A1 | |
| Obtain position vector $3\mathbf{i} + \mathbf{j} + \mathbf{k}$ of intersection point, or equivalent | A1 | The follow through is on $3\mathbf{i} + \mathbf{j} + \mathbf{k}$ only |
**Total: [5]**
---
## Question 10(ii):
| Answer/Working | Mark | Guidance |
|---|---|---|
| **EITHER:** Use scalar product to obtain $2a + b + 3c = 0$ and $a - 2b + c = 0$ | B1 | |
| Solve and find one ratio e.g. $a : b$ | M1 | |
| State one correct ratio | A1 | |
| Obtain answer $a : b : c = 7 : 1 : -5$, or equivalent | A1 | |
| Substitute coordinates of a relevant point and values of $a$, $b$ and $c$ in general equation of plane and calculate $d$ | M1 | |
| Obtain answer $7x + y - 5z = 17$, or equivalent | A1 | |
| **OR:** Using two points on $l$ and one on $m$ (or vice versa) state three simultaneous equations in $a, b, c$ and $d$ e.g. $3a + b + c = d$, $a - 2c = d$ and $6a - 5b + 4c = d$ | B1$\sqrt{}$ | |
| Solve and find one ratio e.g. $a : b$ | M1 | |
| State one correct ratio | A1 | |
| Obtain a ratio of three unknowns e.g. $a : b : c = 7 : 1 : -5$, or equivalent | A1 | |
| Use coordinates of a relevant point and found ratio to find fourth unknown e.g. $d$ | M1 | |
| Obtain answer $7x + y - 5z = 17$, or equivalent | A1 | |
| **OR:** Form a correct 2-parameter equation for the plane, e.g. $\mathbf{r} = \mathbf{i} - 2\mathbf{k} + \lambda(2\mathbf{i}+\mathbf{j}+3\mathbf{k}) + \mu(\mathbf{i}-2\mathbf{j}+\mathbf{k})$ | B1$\sqrt{}$ | |
| State 3 equations in $x, y, z, \lambda$ and $\mu$ | M1 | |
| State 3 correct equations | A1$\sqrt{}$ | |
| Eliminate $\lambda$ and $\mu$ | M1 | |
| Obtain equation in any correct unsimplified form | A1 | |
| Obtain $7x + y - 5z = 17$, or equivalent | A1 | |
| **OR:** Attempt to calculate vector product of vectors parallel to $l$ and $m$ | M1 | |
| Obtain two correct components of the product | A1 | |
| Obtain correct product, e.g. $7\mathbf{i}+\mathbf{j}-5\mathbf{z}$ | A1 | |
| State that the plane has equation of the form $7x + y - 5z = d$ | A1$\sqrt{}$ | |
| Substitute coordinates of a relevant point and calculate $d$ | M1 | |
| Obtain answer $7x + y - 5z = 17$, or equivalent | A1 | |
**Total: [6]**10 The lines $l$ and $m$ have vector equations
$$\mathbf { r } = \mathbf { i } - 2 \mathbf { k } + s ( 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ) \quad \text { and } \quad \mathbf { r } = 6 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } + t ( \mathbf { i } - 2 \mathbf { j } + \mathbf { k } )$$
respectively.\\
(i) Show that $l$ and $m$ intersect, and find the position vector of their point of intersection.\\
(ii) Find the equation of the plane containing $l$ and $m$, giving your answer in the form $a x + b y + c z = d$.
\hfill \mbox{\textit{CAIE P3 2003 Q10 [11]}}