Topic Brief: Position Vectors Distance and Foot of Perpendicular

Provenance: compiled 2026-09-29. Sources checked directly for this document: research/veridian/WMA14-verified-facts.md §1 (verbatim spec text for 7.4–7.5), §5.7 (the full October 2021 Q7 entry — examiner's-report quote, mark-scheme quotes for parts (a)(i) and (b), and the independently-recomputed λ=37/50-vs-2/5 arithmetic), §6 (series/document-count methodology), §7 (this topic is not among those judged to have enough verified material for its own standalone lesson), §8 (flagged gaps), and the dedicated lesson-audit-log entry for this exact lesson file (dated 2026-09-14, which corrected a misidentified trap and added the previously-untaught part-(b) reflection method); content/veridian/curriculum/wma14/position-vectors-distance-and-foot-of-perpendicular.ts (full header/flag disclosure and the marked-solution's own mark codes, checked to confirm which content is the one real citation versus VERIDIAN-original scaffolding — not reproduced here); content/veridian/technique-hub/wma14/codex-deep-dive-position-vectors-distance-and-foot-of-perpendicular.md in full (its §2–3 theory and case taxonomy, and its §4 exam-angle synthesis, are cited below, not restated); signal-frequency-and-boundaries.md §1's existing frequency row for this topic and §2's grade-boundary gap; and forge-exam-technique-guide.md §1's pacing table (the ~11-minute estimate) and §2's mark-type-sequence note. content/veridian/question-bank/wma14.ts was checked directly and contains no entry for this topic. No fresh PDF extraction was performed — every fact below reuses citations already verified elsewhere in this facts bank. Not affiliated with or endorsed by Pearson Edexcel.

#SIGNAL Topic Master Brief — Position Vectors, Distance, and the Foot of a Perpendicular (spec 7.4–7.5)

#Frequency: seen in 1 of 7 series checked

Of the 11 WMA14 series with at least one primary document in this facts bank, 7 received real, question-by-question mining rather than a boilerplate-only check: January 2021, October 2021, June 2022, October 2022, January 2024, October 2024, January 2026 (signal-frequency-and-boundaries.md §1). Exactly one of those seven — October 2021, WMA14/01, Q7 (Publications Code WMA14_01_2110_MS) — carries a verified citation for the foot-of-perpendicular application of spec 7.4–7.5. Read "1 of 7" as this course's own sourcing gap, not a rarity signal: signal-frequency-and-boundaries.md §1 explicitly names this row (alongside area-under-a-parametric-curve) as the thinnest-evidenced content in the whole facts bank, and 7.4–7.5 are compulsory, explicitly spec-listed skills that no primary source anywhere in this course's review shows Pearson omitting by design.

#Mark-tariff pattern: one real question, and a real gap inside it

The confirmed item is 9 marks (Oct 2021 Q7, parts (a)+(b): 7+2). Inside part (a), the facts bank verbatim-quotes only the mark scheme for (a)(i) — finding the foot of the perpendicular, X — as a four-mark chain: M1 (form $\overrightarrow{AX}$ from the line's general point, either order) → M1 (the perpendicularity condition $\overrightarrow{AX}\cdot\mathbf d=0$, then solve) → A1 (the correct parameter value) → dM1, explicitly dependent on the preceding M1, for substituting back to get X's coordinates. That accounts for 4 of the 7 marks in part (a); the remaining 3 marks belong to (a)(ii), the shortest-distance calculation itself, and this facts bank does not capture that sub-part's own M/A breakdown anywhere — a genuine, flagged gap in the sourcing, not a claim that those marks don't exist. Part (b) is a clean, separately-quoted two-mark item: M1 ("Uses AX = XB or similar correct method") + A1 (the correct position vector of B). The dependent-mark structure in (a) — a method mark for substituting a solved parameter back in, gated on the mark before it — is the same dM1 pattern forge-exam-technique-guide.md §2 flags as a recurring feature of this paper's own scheme design, not unique to this topic.

#Context: never a single isolated ask

In its one real appearance, this spec area was not tested as a bare "find the distance" question. The same external point and the same line carried three chained asks in succession — the foot of the perpendicular (a)(i), the shortest distance (a)(ii), and the reflection of the external point in the line (b), the last of which reuses spec 7.4's own position-vector identity a second time via the midpoint relationship $\overrightarrow{AX}=\overrightarrow{XB}$. That is a single real data point, not a confirmed recurring pattern across series — no second sighting exists in this facts bank to check whether every appearance of this topic stacks all three asks, or whether Oct 2021 Q7 happened to.

#What is, and isn't, on record

The examiner's report documents exactly one consequential error for this question: dotting the line's raw general point directly against the direction vector, instead of first forming $\overrightarrow{AX}$ — "This immediately resulted in 0 marks for (a)(i)." This course's own lesson-audit log (facts bank, dated 2026-09-14) corrected an earlier misreading of that trap: the wrong equation is not "the direction vector dotted with itself" (which would be unsolvable), but a genuinely solvable equation that happens to answer a different question — it gives $\lambda=37/50$, a specific plausible-looking wrong value, against the correct $\lambda=2/5$. The same report credits a named alternative for part (a): minimising $|\overrightarrow{AX}|$ directly, "using either differentiation or by completing the square." No part-(b)-specific wrong path is documented — the report notes only that many candidates did not attempt (b) after struggling with (a), not a specific misconception — so nothing here claims a (b)-specific trap. No grade-boundary data exists anywhere in this course's WMA14 source review (signal-frequency-and-boundaries.md §2), so none is implied for this topic either.

#If this topic appears: what to expect

On the one confirmed data point available: do not expect a bare, stand-alone "find the shortest distance" prompt. Expect a foot-of-perpendicular setup to be asked alongside at least one further step on the same line and point — the real item chained the foot, the distance, and a reflection into one nine-mark, roughly 11-exam-minute question (forge-exam-technique-guide.md §1). Form and write down $\overrightarrow{AX}$ as its own visible line before dotting it with the direction vector — that ordering is what the mark scheme's M1 actually rewards, and the coordinates-of-X step that follows is a dependent mark, not an independent one, so an examiner has nothing to credit if the intermediate parameter value isn't shown. If the dot-product route stalls, the examiner's report itself names a legitimate fallback for the foot/distance part: minimise $|\overrightarrow{AX}|^2$ as a quadratic in the parameter, by differentiation or by completing the square. Watch specifically for the one documented trap — dotting the line's own general point against the direction vector, skipping the subtraction that forms $\overrightarrow{AX}$ — because it produces a plausible-looking wrong answer, not an equation that visibly fails. Treat "seen in 1 of 7" as a gap in this course's evidence rather than permission to under-revise the skill — it is required by the spec regardless of how often this review has caught it in the wild.