| Exam Board | SPS |
|---|---|
| Module | SPS SM Pure (SPS SM Pure) |
| Year | 2023 |
| Session | June |
| Marks | 6 |
| Topic | Vectors Introduction & 2D |
| Type | Ratio division of line segment |
| Difficulty | Moderate -0.8 This is a straightforward vectors question testing basic operations: finding magnitude using Pythagoras (part a), using parallelogram properties with vector addition (part b), and section formula/ratio theorem for collinear points (part c). All three parts are standard textbook exercises requiring only routine application of well-practiced techniques with no problem-solving insight needed. |
| Spec | 1.10b Vectors in 3D: i,j,k notation1.10c Magnitude and direction: of vectors1.10f Distance between points: using position vectors |
Relative to a fixed origin $O$,
• the point $A$ has position vector $5\mathbf{i} + 3\mathbf{j} - 2\mathbf{k}$
• the point $B$ has position vector $7\mathbf{i} + \mathbf{j} + 2\mathbf{k}$
• the point $C$ has position vector $4\mathbf{i} + 8\mathbf{j} - 3\mathbf{k}$
(a) Find $|\vec{AB}|$ giving your answer as a simplified surd. [2]
Given that $ABCD$ is a parallelogram,
(b) find the position vector of the point $D$. [2]
The point $E$ is positioned such that
• $ACE$ is a straight line
• $AC : CE = 2 : 1$
(c) Find the coordinates of the point $E$. [2]
\hfill \mbox{\textit{SPS SPS SM Pure 2023 Q4 [6]}}