| Exam Board | OCR |
|---|---|
| Module | C4 (Core Mathematics 4) |
| Year | 2007 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Vectors 3D & Lines |
| Type | Angle between two vectors/lines (direct) |
| Difficulty | Moderate -0.3 This is a straightforward two-part vectors question requiring standard techniques: (i) finding distance using |b-a| and (ii) applying the scalar product formula to find an angle. Both are routine C4 procedures with no conceptual challenges, making it slightly easier than average but not trivial due to the arithmetic involved. |
| Spec | 1.10c Magnitude and direction: of vectors4.04c Scalar product: calculate and use for angles |
| Answer | Marks | Guidance |
|---|---|---|
| (i) Find \(a - b\) or \(b - a\) irrespective of label | M1 | (expect \(11i - 2j - 6k\) or \(-11i + 2j + 6k\)) |
| Method for magnitude of any vector \(\sqrt{161}\) or \(12.7(12.688578)\) | M1 | A1 |
| (ii) Using \(\overrightarrow{AO}\) or \(\overrightarrow{OA}\) and \( | \overrightarrow{AB} | \) or \( |
| \(\cos \theta = \frac{\text{scalar product of any two vectors}}{\text{product of their moduli}}\) | M1 | |
| 43 or better \((42.967...), 0.75\) or better \((0.7499218...\) | A1 | 3 marks |
(i) Find $a - b$ or $b - a$ irrespective of label | M1 | (expect $11i - 2j - 6k$ or $-11i + 2j + 6k$)
Method for magnitude of any vector $\sqrt{161}$ or $12.7(12.688578)$ | M1 | A1 | 3 marks
(ii) Using $\overrightarrow{AO}$ or $\overrightarrow{OA}$ and $|\overrightarrow{AB}|$ or $|\overrightarrow{BA}|$ | B1 | Do not class angle $AOB$ as MR
$\cos \theta = \frac{\text{scalar product of any two vectors}}{\text{product of their moduli}}$ | M1 |
43 or better $(42.967...), 0.75$ or better $(0.7499218...$ | A1 | 3 marks | If 137 obtained, followed by 43, award A0. Common answer 114 probably → B0 M1 A0 |
---The points $A$ and $B$ have position vectors $\mathbf{a}$ and $\mathbf{b}$ relative to an origin $O$, where $\mathbf{a} = 4\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$ and $\mathbf{b} = -7\mathbf{i} + 5\mathbf{j} + 4\mathbf{k}$.
\begin{enumerate}[label=(\roman*)]
\item Find the length of $AB$. [3]
\item Use a scalar product to find angle $OAB$. [3]
\end{enumerate}
\hfill \mbox{\textit{OCR C4 2007 Q3 [6]}}