Deep Dive: Resultant Forces and Bearings

Provenance: written 2026-09-29. Read in full for this pass: research/veridian/WME01-verified-facts.md (the 17-series verified-facts bank — §1 spec text and prerequisites, §2 formula-booklet status, §4 topic 2 in full, §6–§8, and the lesson's own two audit-trail entries at §9, lines 295–316); content/veridian/curriculum/wme01/resultant-forces-and-bearings.ts in full (its own header, mechanism blocks, diagram, method-comparison, both marked-solutions, chain-drill and trap-taxonomy are not reproduced below); all three existing WME01 technique-hub documents in full (codex-formula-and-method-bank.md §1.3/§2.1–§2.2, forge-exam-technique-guide.md §1, signal-frequency-and-boundaries.md §1); the existing codex-deep-dive-equilibrium-of-a-particle-coplanar-forces.md (read for format precedent only, not this topic's content); and content/veridian/question-bank/wme01.ts in full (one item touches this topic's own spec territory — Jan 2025 Q1, tagged 2.2 — cross-checked below to confirm no numeric overlap). No new primary-source PDF was fetched for this pass; every claim traces to one of the documents just named, each of which already traces its own claims to a real spec, mark-scheme, examiner-report or question-paper PDF.

#CODEX Deep-Dive — Resultant Forces and Bearings

#Pearson Edexcel IAL Mechanics 1 (WME01/01), spec 2.1–2.2

This is not a fourth re-teach of the topic. The lesson file owns the full worked chains, the diagram, the method-comparison block, both marked-solutions and the MCQs, and none of that is repeated here. codex-formula-and-method-bank.md already gives this topic a formula masterlist entry and a short method skeleton with its own numeric checks. What none of the four existing documents does is set out, in one place, every real distinct sub-case this exact spec territory actually contains, illustrate each with a short check that is numerically distinct from every check those four documents already own, and assemble the complete, honestly-scoped record of every real exam angle this specific topic has actually been tested from. That is this document's job.

#1. What spec 2.1–2.2 asks for, and what it assumes already exists

Spec 2.1 is two sentences: "Magnitude and direction of a vector. Resultant of vectors may also be required," with guidance that students "may be required to resolve a vector into two components or use a vector diagram," and that questions "may be set involving the unit vectors i and j." Spec 2.2 adds "application of vectors to displacements, velocities, accelerations and forces in a plane." Neither sentence restricts "resultant of vectors" to exactly two vectors — a fact the lesson's own worked material never exercises beyond the two-force case, and §2.4 below closes that gap.

One prerequisite fact worth stating plainly because it is easy to miss: M1's own unit description requires "a knowledge of P1 and P2 and associated formulae and of vectors in two dimensions" as a starting condition, not a topic this unit teaches from zero. Bearings, i/j notation, and basic vector addition are assumed entry knowledge; what M1 actually adds is the mechanical-physics content — treating a bearing/magnitude pair as a force, and finding a resultant force rather than an abstract resultant vector.

The formula-booklet status matters for how each sub-case below should actually be revised: the cosine rule (§2.2) is a genuine Pure Mathematics P1/P2 result the booklet extends to M1 candidates and is confirmed as live-tested (Jun 2024 Q2) — it can legitimately be looked up. Lami's theorem (§2.3) is not a booklet entry at all, despite being confirmed examiner-credited in two series — it must be recalled or derived from the sine rule, not looked up on exam day. Whether the sine rule itself sits inside the P1/P2 pages available to M1 candidates is not independently confirmed anywhere in this course's research; codex-formula-and-method-bank.md already flags this exact gap and it is not resolved here either.

#2. The complete real model, sub-case by sub-case

#2.1 Resultant of two forces — resolve into components, then Pythagoras

Mechanism. Convert each force to i/j form, add the i-components, add the j-components, then apply Pythagoras to the combined pair. This needs no angle identified between the two forces at all, because the i- and j-axes are exactly perpendicular by construction — which is precisely why a real examiner report records this route as the more successful of the two available (Jun 2024, Q2, quoted in full at §3 below).

Micro-check (distinct from every number the lesson, the diagram, and codex-formula-and-method-bank.md's own §2.1 check use): forces P = 7 N on bearing 040° and Q = 11 N on bearing 160° act at a point. P resolves to 4.50i + 5.36j (N); Q resolves to 3.76i − 10.34j (N). Resultant = 8.26i − 4.97j (N), magnitude √93.0 ≈ 9.64 N (3 s.f.), on bearing 121° (nearest degree) — the negative j-component and positive i-component place the answer correctly in the south-east quadrant before any arithmetic is trusted.

#2.2 Resultant of two forces — triangle of forces / cosine rule

Mechanism. Draw the two forces head-to-tail as two sides of a triangle; the third side is the resultant. The angle the cosine rule needs is the triangle's own interior angle at the shared vertex, which is the supplement of the angle between the forces as drawn from their common point of application, never that angle itself — because translating one force to start where the other ends does not rotate it; one arm of the original angle is effectively reversed.

Cross-check, same scenario as §2.1. The angle between P and Q as drawn from their common point is 160° − 40° = 120°. Using $R^2 = P^2+Q^2+2PQ\cos\theta$ with the raw 120° (equivalent to using the triangle's own interior angle of 60° with a minus sign, since $\cos(180°-\theta)=-\cos\theta$): $R^2 = 49+121+2(7)(11)\cos120° = 170 - 77 = 93$, so $R = \sqrt{93} \approx 9.64$ N — identical to §2.1's answer, confirming both routes describe the same geometry. The real, verified trap this route carries: an examiner report on exactly this comparison records candidates using "an incorrect angle of 30, 60 or in some cases 330 degrees, even when they had drawn what appeared to be the correct obtuse angled triangle" (Jun 2024, Q2) — already the lesson's own central trap-taxonomy item and not re-derived further here.

#2.3 Three concurrent coplanar forces in equilibrium — Lami's theorem

Mechanism. For three forces P, Q, T concurrent at a point and in equilibrium, $\dfrac{P}{\sin\alpha}=\dfrac{Q}{\sin\beta}=\dfrac{T}{\sin\gamma}$, where each angle is the one between the other two forces, and the three angles always sum to 360°. Confirmed as a genuinely examiner-credited alternative to resolving in two independent series (Jan 2022 Q1: "another direct method involved applying Lami's theorem"; Jan 2023 Q6: "an alternative method was to consider a triangle of forces and use the sine and/or cosine rule (or Lami's Theorem)") — despite being named nowhere in the spec's own wording.

Micro-check (distinct from the lesson's own reference-card statement of the formula, which carries no worked numbers): three coplanar forces are in equilibrium with the angle between Q and T equal to 110°, between P and T equal to 130°, and between P and Q equal to 120° — these three sum to exactly 360°, a check worth running before trusting any of the three numbers below. Taking P = 10 N: $\dfrac{10}{\sin110°} = 10.64$, so $Q = 10.64\sin130° \approx 8.15$ N and $T = 10.64\sin120° \approx 9.22$ N.

#2.4 Resultant of three or more forces — the general component-sum case

Mechanism. Spec 2.1's own wording ("resultant of vectors," not "of two vectors") covers any number of concurrent forces, and the method is the direct extension of §2.1: sum every i-component across all the forces, sum every j-component separately, then take the magnitude/direction of the single combined vector. No triangle-of-forces or Lami's-theorem route generalises past three forces the way this one does.

Micro-check (distinct from codex-formula-and-method-bank.md's own two-force check, P = 6i − 2j N and Q = −i + 5j N → 5i + 3j N): three forces $F_1=(3\mathbf{i}+5\mathbf{j})$ N, $F_2=(-7\mathbf{i}+2\mathbf{j})$ N, $F_3=(4\mathbf{i}-9\mathbf{j})$ N act at a point. Resultant $=(3-7+4)\mathbf{i}+(5+2-9)\mathbf{j}=-2\mathbf{j}$ N — exactly 2 N due south (bearing 180°), the i-components happening to cancel completely.

Real exam anchor for this exact mechanic. January 2025 Q1(a) (spec territory tagged 2.1/4.1 by this paper's own FORGE guide) gives a particle three real named forces and asks for their resultant before a follow-up part; per that question's own mark scheme, summing them gives $-12\mathbf{i}+18\mathbf{j}$ N (quoted from the mark-scheme working already transcribed in content/veridian/question-bank/wme01.ts, not reproduced here as a full worked exemplar — that file, and the sibling lesson newtons-second-law-in-vector-form.ts, already own it). This confirms the three-force generalisation above is not a hypothetical extension of spec 2.1's wording; it is the literal opening step of a real, recently-examined question.

#2.5 Bearing → component conversion

Mechanism. A bearing is measured clockwise from north, the reverse starting axis from a standard angle measured from east — which swaps the usual pairing: for a force of magnitude F on bearing θ, the east/i-component is $F\sin\theta$ and the north/j-component is $F\cos\theta$. This is the single most-confirmed vectors-specific trap in the whole archive; the mechanism derivation and the full reconstructed wrong-answer marked-solution both already belong to the lesson and are not repeated here.

Micro-check (distinct from every bearing the lesson, the diagram, the chain-drill and codex-formula-and-method-bank.md's own §2.2 check use — 030°, 053°, 070°, 120°, 200°, 065° respectively): a force of 15 N on bearing 245° resolves to $15\sin245°\mathbf{i}+15\cos245°\mathbf{j} \approx -13.6\mathbf{i}-6.34\mathbf{j}$ (N) — both components negative, consistent with 245° sitting in the south-west quadrant, between south (180°) and west (270°).

#2.6 Position vector at time t, then distance from a fixed point

Mechanism. For a particle with constant velocity, $\mathbf{r}(t)=\mathbf{r}_0+t\mathbf{v}$; distance from a stated fixed point A is Pythagoras applied to the difference $\mathbf{r}(t)-\mathbf{r}_A$, never to $\mathbf{r}(t)$ alone (which gives distance from the origin, almost never the point actually asked about). This sub-case was added to the lesson itself only in a 2026-09-25 pass, on the strength of a second corroborating series (Jan 2025 Q5(b)) after an earlier pass had left it as a single-series, not-lesson-worthy note (Jun 2024 Q7). The full mechanism block and marked-solution are the lesson's own and are not repeated here.

Micro-check (distinct from the lesson's own scenario — $\mathbf{r}_0=\mathbf{i}-2\mathbf{j}$ m, $\mathbf{v}=4\mathbf{i}+3\mathbf{j}$ m/s, fixed point $17\mathbf{i}+10\mathbf{j}$ m, $t=5$ s, distance 5 m exactly): a particle has $\mathbf{r}_0=(2\mathbf{i}+3\mathbf{j})$ m and constant velocity $(3\mathbf{i}-2\mathbf{j})$ m/s. At $t=6$ s, $\mathbf{r}=(20\mathbf{i}-9\mathbf{j})$ m. Its distance from the fixed point $B=(17\mathbf{i}-11\mathbf{j})$ m: difference $=(3\mathbf{i}+2\mathbf{j})$ m, distance $=\sqrt{13}\approx3.61$ m (3 s.f.).

#2.7 Relative position between two moving particles — minimum separation

Mechanism, and where the real evidence actually stops. The same Jun 2024 Q7 that supplies §2.5's bearing trap continues into a harder, single-series-confirmed variant: build the relative position vector $\mathbf{AB}(t)$ between two moving particles, then find their minimum separation by differentiating $|\mathbf{AB}(t)|^2$ and setting the derivative to zero. The verified-facts bank records two real, distinct errors on this exact continuation: candidates using "the derivative method often confused the t value, when the minimum occurs (7/12), with the minimum value of $d^2$ (80)" — conflating when the minimum occurs with what it is — and, separately, on what the facts bank describes as "the alternative equal-components method," "a significant number of candidates wrote down an incorrect time of 24/10 rather than 10/24." The differentiation mechanism is confidently reconstructable and is worked below; the second, "equal-components" method's exact mechanics are not — the facts bank's own description of it is a paraphrase, not a verbatim examiner-report sentence, and no primary-source text recovered for this course states precisely what is being equated to what. That gap is recorded here rather than guessed at.

Micro-check (VERIDIAN-original, illustrating only the differentiation mechanism and the verified t-value/minimum-value conflation — not a reconstruction of the real question's own particles, velocities, or numbers, which were not recovered): particles A and B have position vectors $\mathbf{r}_A(t)=(1+2t)\mathbf{i}+(3+t)\mathbf{j}$ and $\mathbf{r}_B(t)=(7-t)\mathbf{i}+2t\mathbf{j}$. Their relative position $\mathbf{AB}=\mathbf{r}_B-\mathbf{r}_A=(6-3t)\mathbf{i}+(t-3)\mathbf{j}$, so $|\mathbf{AB}|^2=(6-3t)^2+(t-3)^2=10t^2-42t+45$. Differentiating and setting to zero: $20t-42=0 \Rightarrow t=2.1$ — the answer to when the particles are closest. Substituting back, $|\mathbf{AB}|^2=10(2.1)^2-42(2.1)+45=0.9$, so the minimum distance is $\sqrt{0.9}\approx0.949$ m (3 s.f.) — the answer to what the closest distance is. The positive coefficient of $t^2$ (10) confirms the parabola opens upward, so the stationary point found is genuinely a minimum, not a maximum — the same cross-check discipline this lesson's own marked-solutions apply throughout. $t=2.1$ and the minimum distance $\approx0.949$ m are two different numbers answering two different questions; treating either as the other's answer is exactly the real, verified error named above.

#3. Every real exam angle this specific topic has actually been tested from

Frequency, honestly scoped. In the two most recent series this course has fully mark-mapped question-by-question, spec 2.1–2.2 anchored real marks in both — Jan 2025 (Q1, Q5) and Jun 2025 (Q2, Q7) — 2 of 2 series checked this way (signal-frequency-and-boundaries.md §1). The bearing-swap trap and the wrong-angle-in-the-cosine-rule trap are each currently confirmed in exactly 1 of the 17 series this course's archive has reviewed (both Jun 2024); Lami's theorem's credited status is confirmed independently in 2 of those 17 series (Jan 2022, Jan 2023). No percentage and no likelihood claim is made anywhere above — each number is a count of series actually read.

One further provenance note, recorded but not built into a worked example, matching this course's own standing convention against reproducing a real past paper's own figures verbatim: a later audit pass recovered Jun 2024 Q2's own real force magnitudes and angle for the first time — P = 10 N due west, Q = 8 N on bearing 330°, 60° between them — giving $F=\sqrt{244}=2\sqrt{61}\approx15.6$ N. This is newly confirmable arithmetic ($100+64+2(10)(8)\cos60°=164+80=244$), not a new teaching example; the lesson's own VERIDIAN-original scenario (8 N/5 N/70°) was deliberately left as-is.

#4. What remains unconfirmed

The "equal-components" alternative method for the minimum-separation variant (§2.7) is named in the facts bank but its exact mechanics were never recovered from a primary source checked for this course. Whether the sine rule itself sits inside the P1/P2 formula-booklet pages extended to M1 candidates is unverified, as codex-formula-and-method-bank.md already flags. Jun 2025 Q2 and Q7 are confirmed by spec-tag only, not by mark-scheme or examiner-report content, so no trap, mark split, or method preference can be claimed for either beyond the fact that they exist and sit in this topic's territory. No topic-specific mark-weighting table exists anywhere in this course's WME01 research (facts bank §8's own flag); the frequency claims above are series-counts, never proportions of a paper's marks.

#Sources

• research/veridian/WME01-verified-facts.md — spec text and prerequisites (§1), formula-booklet status (§2), topic 2 in full (§4), topics list (§6), flagged gaps (§8), and the resultant-forces-and-bearings lesson's own two audit-trail entries (§9, lines 295–316), itself sourced to WME01_01_2406 (Jun 2024 Q2, Q7), WME01_01_2201 (Jan 2022 Q1), WME01_01_2301 (Jan 2023 Q6), and WME01_01_2501 (Jan 2025 Q1, Q5(b)) mark schemes and examiner reports.
• content/veridian/curriculum/wme01/resultant-forces-and-bearings.ts — the lesson this document deliberately does not duplicate.
• content/veridian/technique-hub/wme01/codex-formula-and-method-bank.md §1.3, §2.1–§2.2 — formula status and method skeletons for this spec point, cross-checked to keep every micro-example above numerically distinct from its own checks.
• content/veridian/technique-hub/wme01/forge-exam-technique-guide.md §1 — the two fully mark-mapped series' own question-by-question spec tags, used for §3's Jan/Jun 2025 rows.
• content/veridian/technique-hub/wme01/signal-frequency-and-boundaries.md §1 — the "2 of 2 series" frequency finding and the cross-series trap counts, both reused rather than re-derived.
• content/veridian/question-bank/wme01.ts — Jan 2025 Q1(a)–(b)'s real mark-scheme working, cited for §2.4's anchor fact only, not reproduced as a worked exemplar.