| Exam Board | SPS |
|---|---|
| Module | SPS FM (SPS FM) |
| Year | 2022 |
| Session | January |
| Marks | 7 |
| Topic | Vectors 3D & Lines |
| Type | Foot of perpendicular from origin to line |
| Difficulty | Standard +0.3 This is a standard Further Maths vectors question requiring routine techniques: finding a direction vector from two points, writing a line equation, then using the perpendicularity condition (dot product = 0) to find a parameter value. While it involves multiple steps, each is a textbook procedure with no novel insight required, making it slightly easier than average. |
| Spec | 1.10a Vectors in 2D: i,j notation and column vectors1.10b Vectors in 3D: i,j,k notation1.10d Vector operations: addition and scalar multiplication4.04a Line equations: 2D and 3D, cartesian and vector forms4.04c Scalar product: calculate and use for angles |
3.
Relative to an origin $O$, the points $A$ and $B$ have position vectors $3 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k }$ and $\mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k }$ respectively.\\
(i) Find a vector equation of the line passing through $A$ and $B$.\\
(ii) Find the position vector of the point $P$ on $A B$ such that $O P$ is perpendicular to $A B$.\\[0pt]
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\hfill \mbox{\textit{SPS SPS FM 2022 Q3 [7]}}