| Exam Board | CAIE |
|---|---|
| Module | P1 (Pure Mathematics 1) |
| Year | 2009 |
| Session | June |
| Marks | 7 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors: Lines & Planes |
| Type | Vector operations and magnitudes |
| Difficulty | Moderate -0.3 This is a straightforward vector question requiring standard operations: dot product calculation, angle determination from sign of dot product, position vector arithmetic, and unit vector calculation. All techniques are routine A-level procedures with no problem-solving insight needed, making it slightly easier than average. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication1.10e Position vectors: and displacement |
| Answer | Marks | Guidance |
|---|---|---|
| (i) \(\vec{OA} \cdot \vec{OB} = 14 - 16 - 4 = -6\) | M1 A1 | Must be scalar from correct method |
| This is \(-ve \rightarrow\) Obtuse angle | B1√ [3] | co. Correct deduction from his scalar |
| (ii) \(\vec{AB} = 5\mathbf{i} + 10\mathbf{j} - 5\mathbf{k}\) | Needs \(\vec{AB}\) and \(\vec{OX}\) attempting | |
| \(\vec{AX} = \frac{2}{5}(\vec{AB})\) | M1 | |
| \(\vec{OX} = \vec{OA} + \vec{AX}\) | ||
| \(\vec{OX} = 4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}\) | A1 | |
| Divides by the modulus | M1 | Must finish with a vector, not a scalar |
| Unit vector \(= \frac{1}{6}(4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})\) | A1√ [4] | Correct for his \(\vec{OX}\) |
(i) $\vec{OA} \cdot \vec{OB} = 14 - 16 - 4 = -6$ | M1 A1 | Must be scalar from correct method
This is $-ve \rightarrow$ Obtuse angle | B1√ [3] | co. Correct deduction from his scalar
(ii) $\vec{AB} = 5\mathbf{i} + 10\mathbf{j} - 5\mathbf{k}$ | | Needs $\vec{AB}$ and $\vec{OX}$ attempting
$\vec{AX} = \frac{2}{5}(\vec{AB})$ | M1 |
$\vec{OX} = \vec{OA} + \vec{AX}$ | |
$\vec{OX} = 4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k}$ | A1 |
| |
Divides by the modulus | M1 | Must finish with a vector, not a scalar
Unit vector $= \frac{1}{6}(4\mathbf{i} - 4\mathbf{j} + 2\mathbf{k})$ | A1√ [4] | Correct for his $\vec{OX}$6 Relative to an origin $O$, the position vectors of the points $A$ and $B$ are given by
$$\overrightarrow { O A } = 2 \mathbf { i } - 8 \mathbf { j } + 4 \mathbf { k } \quad \text { and } \quad \overrightarrow { O B } = 7 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }$$
(i) Find the value of $\overrightarrow { O A } \cdot \overrightarrow { O B }$ and hence state whether angle $A O B$ is acute, obtuse or a right angle.\\
(ii) The point $X$ is such that $\overrightarrow { A X } = \frac { 2 } { 5 } \overrightarrow { A B }$. Find the unit vector in the direction of $O X$.
\hfill \mbox{\textit{CAIE P1 2009 Q6 [7]}}