| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2012 |
| Session | June |
| Marks | 5 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors Introduction & 2D |
| Type | Angle between two vectors |
| Difficulty | Moderate -0.8 This is a straightforward application of the dot product formula for angles and the magnitude formula. Part (i) requires direct substitution into cos θ = a·b/(|a||b|), and part (ii) uses |a-b|² = |a|² + |b|² - 2a·b. Both are standard textbook exercises with all values given, requiring only routine recall and calculation. |
| Spec | 1.10c Magnitude and direction: of vectors1.10d Vector operations: addition and scalar multiplication |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Use \(\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{ | \mathbf{a} | |
| Obtain \(\left(\cos\theta = \frac{6}{12}\right)\), \(\theta = 60°\) or \(\frac{1}{3}\pi\) or 1.05 or better | A1 | Better: 1.0471976 (rot) |
| Total: [2] |
| Answer | Marks | Guidance |
|---|---|---|
| Answer | Marks | Guidance |
| Indicate \(\mathbf{a}-\mathbf{b}\) is vector joining ends of \(\mathbf{a}\) and \(\mathbf{b}\) or equiv; \( | \mathbf{a}-\mathbf{b} | = |
| Use cosine rule correctly on 3, 4 and included angle from (i) | M1 | Or any other correct method |
| Obtain \(\sqrt{13}\) or 3.61 or better (No ft from wrong \(\theta\)) | A1 | 3.6055513 (rot) |
| Total: [3] |
# Question 5:
## Part (i)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Use $\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}||\mathbf{b}|}$ | M1 | |
| Obtain $\left(\cos\theta = \frac{6}{12}\right)$, $\theta = 60°$ or $\frac{1}{3}\pi$ or 1.05 or better | A1 | Better: 1.0471976 (rot) |
| **Total: [2]** | | |
## Part (ii)
| Answer | Marks | Guidance |
|--------|-------|----------|
| Indicate $\mathbf{a}-\mathbf{b}$ is vector joining ends of $\mathbf{a}$ and $\mathbf{b}$ or equiv; $|\mathbf{a}-\mathbf{b}|=|\mathbf{a}|-|\mathbf{b}|$ or anything similar $\to$ M0 | M1 | |
| Use cosine rule correctly on 3, 4 and included angle from (i) | M1 | Or any other correct method |
| Obtain $\sqrt{13}$ or 3.61 or better (No ft from wrong $\theta$) | A1 | 3.6055513 (rot) |
| **Total: [3]** | | |
---5\\
\includegraphics[max width=\textwidth, alt={}, center]{b5d85e48-0d5a-4edf-bf58-eba4f8d28d3d-2_425_680_1302_689}
In the diagram the points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ with respect to the origin $O$. Given that $| \mathbf { a } | = 3 , | \mathbf { b } | = 4$ and $\mathbf { a . b } = 6$, find\\
(i) the angle $A O B$,\\
(ii) $| \mathbf { a } - \mathbf { b } |$.
\hfill \mbox{\textit{OCR C4 2012 Q5 [5]}}