GAM · Vector Calculations — Strategy & Tricks

📐 The rules the passage assumes you know

A vector is a list of numbers with a direction attached: (x, y) in the plane, (x, y, z) in space. dMAT passages spell out the vectors and restate the formula they want you to use — treat that restatement as a cue. The arithmetic is on you.

a + b = (a₁+b₁, a₂+b₂, a₃+b₃) — add column by column
a − b = (a₁−b₁, a₂−b₂, a₃−b₃) — from A to B is B − A
k·a = (ka₁, ka₂, ka₃) — scaling; k < 0 flips the direction
|a| = √(a₁² + a₂² + a₃²) — length; never the component sum
a·b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b| cos θ — a number, not a vector
|a × b| = |a||b| sin θ — parallelogram area; a × b ⊥ both
û = a/|a| — same direction, length 1
cos θ = a·b / (|a||b|)
a ⊥ b ⇔ a·b = 0;  a ∥ b ⇔ b = k·a — one ratio, every component

Two facts do most of the heavy lifting. First, the dot product is a scalar: if the stem asks for it, every vector option is dead before you compute. Second, both products are bounded — neither |a·b| nor |a × b| can exceed |a||b|, because cos and sin never exceed 1. No other check kills as many distractors.

👀 Name the variant in five seconds

Before you touch a number, classify the question. Six variants cover almost everything, each with exactly one operation attached.

Two extra labels: area, torque, ‘perpendicular to both’ means cross product, usually just |a||b| sin θ; ‘find k such that…’ means turning the stated condition into one equation. If a stem fits no row, re-read the passage — it has defined the operation for you.

🧠 The components table

Every question here is solved the same way: write the vectors as rows, one column per axis, and work down the columns. It looks childish. It is also why you stop dropping minus signs.

Sum the products row: r·u = −31, negative, so the angle between r and u is obtuse. |r + u| = √(16 + 9) = 5, a 3-4-5 triple — no squaring needed. One table, four answers.

1. Extract. Copy every vector from the passage into the table before reading the questions. One passage feeds four or five of them.
2. Label the operation from the stem and write it as a new row.
3. Work the columns one at a time — sign first, then size.
4. Collapse. Magnitude squares-and-adds: |a| = √(x² + y² + z²). Dot multiplies-and-adds: a·b = a₁b₁ + a₂b₂ + a₃b₃, and cos θ = a·b ÷ (|a||b|). Each turns a row into one number.
5. Bound check. At most |a||b|? At least the largest component? Then pick the option.

✂️ Shortcuts that are actually faster

• Pythagorean triples, memorised: 3-4-5, 5-12-13, 8-15-17, 7-24-25 and multiples (6-8-10, 9-12-15). In 3D: (1, 2, 2) → 3, (2, 3, 6) → 7, (1, 4, 8) → 9, (2, 6, 9) → 11.
• The sign of the dot product is free information: positive → acute, zero → right angle, negative → obtuse. Half the angle questions ask no more.
• Cos from memory: 0° → 1, 30° → √3/2, 45° → 1/√2, 60° → 1/2, 90° → 0; sin is the same list backwards. Read it in reverse too: cos θ = 1/2 means 60°.
• Bound before you compute: |a·b| and |a × b| cannot exceed |a||b|. With |a| = 4 and |b| = 5, any option above 20 is dead.
• A magnitude is never smaller than its biggest component: |(3, k, 6)| cannot be 5.
• Square roots multiply cleanly: √2 · √2 = 2, √3 · √12 = 6. Keep them symbolic; approximating √2 ≈ 1.41 mid-way loses the cancellation.
• Parallel by ratio, not by angle: (2, −4, 6) and (−1, 2, −3)? Every component is −2 times the other — parallel and opposite. No cos needed.
• Unit vectors usually hide a triple: (0.6, 0.8), (1/3)(1, 2, 2). If a ‘unit’ vector isn’t unit, make one: divide by the magnitude.

⚡ The 90-second budget

Ninety seconds a question, but a vector pack is four or five questions on one passage — think six or seven minutes for the pack. Spend the first 40 seconds on the table and each question then costs 20–60 seconds.

1. 0–10 s: read the passage once, marking every vector and formula it states.
2. 10–40 s: fill the components table, plus a spare row for the obvious next operation.
3. Per question: 5 s to classify the variant, 10 s to bound-check the options, the rest computing down the columns, 5 s to sanity-check — then commit.

🪤 Seven ways to get it wrong fast

1. Wrong subtraction order. ‘From A to B’ is B − A; the correction returning from r′ to r is r − r′ = −u. An answer with every sign flipped means you subtracted backwards — and that option is always on the list.
2. Forgetting the square root. 6² + 8² = 100 is not a length. Watch for ‘100 m’ next to ‘10 m’.
3. The component sum posing as a magnitude. (6, 8) is not 14 long, yet x + y appears in nearly every magnitude question.
4. A product bigger than |a||b|. sin θ ≤ 1 and cos θ ≤ 1, so neither |a × b| nor |a·b| can exceed |a||b|.
5. Adding magnitudes instead of vectors. |(9, 12)| + |(−3, −4)| = 20, but the sum (6, 8) is only 10 long. Lengths add only when both vectors point the same way.
6. Degrees and radians. π/3 is 60°, cos = 1/2. Read the unit before you read the table.
7. Sin for cos. Dot uses cos (the ‘along’ part), cross uses sin (the ‘across’ part). The 30° torque question offers √3/2 as bait.

🧩 Worked examples, easy to hard

Each example uses a mini passage in the exam’s style. Cover the steps, attempt it inside the time target, then compare.

🎯 Recognition drills

Say the operation out loud, then the answer. No paper on the first pass — the point is to make classification reflexive.

🧭 Guided walk-through: one pack, five questions

Here is a full pack the way it lands in the exam, with the thinking that should run at each moment.

1. Seconds 0–30, the table. Rows r = (7, −2) and u = (−3, 5). Pre-compute r + u = (4, 3) and, because ‘correction’ is in the passage, −u = (3, −5).
2. Q1: r′? Lookup: (4, 3). Five seconds.
3. Q2: the correction returning the robot from r′ to r? r − r′ = −u = (3, −5). The (−3, 5) option is the wrong-order trap. Ten seconds.
4. Q3: final displacement if c = (2, −1)? r′ + c = (6, 2). Ten seconds.
5. Q4: |r′|? (4, 3) → 5 m on sight. A √-something option means the wrong row got squared: |r| = √53, |u| = √34. Five seconds.
6. Q5: angle between r′ and u, acute or obtuse? 4·(−3) + 3·5 = 3 > 0, so acute. Fifteen seconds.

Five questions, about 75 seconds of computing, most of it in the table. Front-load, then harvest.

📈 Progressive practice

Untimed. The goal is zero careless errors, not speed. After each set, sort every mistake into one of the seven traps — the sorting is the practice.

Easy · resultants, magnitudes, perpendicularity → 6 real easy questions · free, no login

Medium · angles, cross-product magnitudes, unknown components → 6 real medium questions · free, no login

Hard · projections, two-condition constraints, 3D packs → 6 real hard questions · free, no login

Move up a tier when you score five of six with the three-question check applied.

⏱️ Speed training

Now the clock: three rounds, the target dropping each time. Write the table every round, even when you think you don’t need it — the habit is what gets faster, not your arithmetic.

1. Round 1 — 75 s per question. Table, classify, bound check, compute, one written sanity check.
2. Round 2 — 60 s. Same routine, sanity check in your head.
3. Round 3 — 45 s. Skip the squaring whenever a triple is visible; guess and move at 45 s.

Timed · medium, 60 s each → 6 real medium questions, against the clock · free, no login

Timed · hard, 75 s each → 6 real hard questions, against the clock · free, no login

🏆 On exam day

Vector packs arrive as one passage plus four or five linked questions — the most predictable points in the GAM, if you run the routine.

• Read the passage once; build the table before question 1.
• Copy the passage’s formula exactly — it decides components versus |a||b| cos θ.
• Classify each stem in five seconds.
• Bound-check the options before computing.
• Subtraction order: ‘from A to B’ is B − A.
• Magnitude: root taken? At least the largest component?
• Sign of the dot product → acute or obtuse; sin for cross, cos for dot.
• Never leave a blank.

Exam simulation · hard passage questions, 90 s each → 6 real hard questions, against the clock · free, no login

Linked questions share a passage, not their answers. If Q3 feels wrong, don’t rebuild Q1 — re-check the single row you used.

📋 Cheat sheet

Add / subtract column by column. From A to B = B − A.
Scale: k·(x, y, z) = (kx, ky, kz); |k·a| = |k||a|.
Magnitude √(x² + y² + z²): take the root; never below the largest component.
Triples: 3-4-5, 5-12-13, 8-15-17, 7-24-25; 3D (1,2,2)→3, (2,3,6)→7, (1,4,8)→9.
Dot: a·b = Σaᵢbᵢ = |a||b| cos θ — a number; |a·b| ≤ |a||b|.
Angle: cos θ = a·b/(|a||b|); sign → acute (+), right (0), obtuse (−).
cos 0°, 30°, 45°, 60°, 90° = 1, √3/2, 1/√2, 1/2, 0; sin is the list backwards.
Perpendicular ⇔ a·b = 0. Parallel ⇔ b = k·a.
Cross: |a × b| = |a||b| sin θ ≤ |a||b|; ⊥ both; parallelogram area.
Unit vector û = a/|a|. Component along û = a·û. Work = F·d.

• Table built before Q1
• Operation named from the stem
• Impossible options crossed out
• Order, root, bound checked
• Answered — never blank