| Exam Board | Pre-U |
|---|---|
| Module | Pre-U 9794/1 (Pre-U Mathematics Paper 1) |
| Year | 2012 |
| Session | Specimen |
| Marks | 6 |
| Topic | Vectors 3D & Lines |
| Type | Angle between two vectors/lines (direct) |
| Difficulty | Moderate -0.3 This is a straightforward two-part question requiring standard vector techniques: finding magnitude of AB using the distance formula, then applying the scalar product formula to find an angle. Both are routine A-level procedures with no conceptual challenges, making it slightly easier than average but not trivial since it requires careful calculation with 3D vectors. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors |
**(i)** Find $\mathbf{a} - \mathbf{b}$ or $\mathbf{b} - \mathbf{a}$ **M1**
Use correct method to find the magnitude of any vector **M1**
$\sqrt{154}$ or equivalent **A1**
**(ii)** Using ($\overrightarrow{AO}$ or $\overrightarrow{OA}$) and ($\overrightarrow{AB}$ or $\overrightarrow{BA}$) **B1**
$\cos\theta = \dfrac{\text{scalar product of any two vectors}}{\text{product of their moduli}}$ **M1**
$32.8°$ or better, or $0.572$ rad or better **A1**
**Total: 6 marks**10 The points $A$ and $B$ have position vectors $\mathbf { a }$ and $\mathbf { b }$ relative to an origin $O$, where $\mathbf { a } = 5 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k }$ and $\mathbf { b } = - 7 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$.\\
(i) Find the length of $A B$.\\
(ii) Use a scalar product to find angle $O A B$.
\hfill \mbox{\textit{Pre-U Pre-U 9794/1 2012 Q10 [6]}}