Module 1: Mathematical Foundations · Theory · assumes Foundation 3
Combining vectors, and taking them apart — from geometric construction to arithmetic you can trust under exam pressure.
Mahavakya
Vectors do not add like numbers, because direction is not a label attached to the quantity — it is part of the quantity itself.
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Learning objectives
After this reading you should be able to:
- Find the magnitude and direction of a resultant using the triangle and parallelogram laws.
- Resolve a vector into perpendicular components along axes you choose, not only horizontal and vertical.
- Add three or more vectors by components, including the quadrant check on the final angle.
- Construct relative velocity and displacement using vector subtraction.
- State the condition under which perpendicular components evolve independently — and when it fails.
Notation. Vectors in bold: A, B, R. Magnitudes in italics or bars: A, |A|. Unit vectors carry a hat: î, ĵ, k̂.
1. The geometry of addition
Two scalars of the same kind combine by arithmetic and nothing else: 3 kg with 4 kg gives 7 kg, whatever the circumstances. Two vectors do not, because the angle between them is part of the question.
Take forces of 3 N and 4 N acting at the same point. Three cases show the whole range:
- Parallel (θ = 0°): the magnitudes add. R = 3 + 4 = 7 N.
- Anti-parallel (θ = 180°): the magnitudes subtract. R = |3 − 4| = 1 N.
- Perpendicular (θ = 90°): Pythagoras applies. R = √(3² + 4²) = 5 N.
Every other angle lies between those extremes, which gives a bound worth committing to memory:
|A − B| ≤ R ≤ A + B
This is worth more than it looks. It is the fastest sanity check available: if a calculation hands you a resultant of 8 N from forces of 3 N and 4 N, the calculation is wrong, and you know it before checking a single step.
2. Triangle and parallelogram laws
Two equivalent constructions combine vectors geometrically. They give the same resultant; which you use is a matter of what you are drawing.
Two operations that undo each other: building one vector from two, and splitting one into two.
Triangle law — tip to tail
Put the tail of B at the tip of A. The resultant runs from the tail of A to the tip of B. This is the one to use for three or more vectors — just keep chaining.
Parallelogram law — tail to tail
Put both tails at one point and complete the parallelogram. The diagonal from that point is the resultant. This is the one to use when both vectors physically act at the same place, as forces on a body do.
Applying the law of cosines to the triangle gives the magnitude. One point causes more confusion than any other here: the law of cosines carries a minus sign, yet the formula below has a plus. The reason is that the interior angle of the construction triangle is 180° − θ, not θ, and cos(180° − θ) = −cos θ. The two minus signs cancel.
R = √A² + B² + 2AB cos θ
Magnitude of the resultant, θ measured between the two vectors
tan α = (B sin θ) / (A + B cos θ)
Direction of the resultant, measured from A
The equal-magnitude case. When A = B, the formula collapses to R = 2A cos(θ/2), and the resultant bisects the angle between them. This appears constantly in problems involving symmetric pairs of forces, and recognising it saves a page of algebra.
3. Resolution
Resolution runs addition backwards. Instead of combining two vectors into one, it replaces one vector with two perpendicular vectors that together produce exactly the same physical effect.
Ax = A cos θ Ay = A sin θ
A = (A cos θ) î + (A sin θ) ĵ
θ measured from the x-axis toward A
Nothing requires those axes to be horizontal and vertical. For a block on a plane inclined at φ, tilting the axes to lie along and across the slope turns a two-dimensional problem into two one-dimensional ones: the weight resolves into mg sin φ down the slope and mg cos φ into it, and the normal force has no component along the slope at all. The physics is identical either way. The algebra is not.
Contemplation
Once a vector is resolved, the two components stop talking to each other. Fire a projectile and the horizontal motion carries on as though gravity did not exist, while the vertical motion behaves as though the object were dropped from rest. Two separate one-dimensional problems, solved apart and rejoined at the end — and the answer is exactly right. It is easy to take this as a fact about forces, that forces simply act independently in nature. It is not. It is a fact about the equations: Newton’s second law is linear, so the x-equation and the y-equation share no terms and can be separated. Add air resistance, where the drag depends on the total speed, and the two motions begin to influence each other immediately. Real projectiles do not decompose so cleanly. Idealised ones do — and every problem you will meet this year is idealised.
4. Addition by components
With three or more vectors, applying the cosine formula repeatedly is slow and error-prone. Resolving everything first turns the whole problem into ordinary arithmetic.
The component method
- Resolve every vector: Ax = A cos θ, Ay = A sin θ. Watch the signs.
- Add the x-components: Rx = ΣAx.
- Add the y-components: Ry = ΣAy.
- Rebuild: R = √(Rx² + Ry²), and θR = tan−1|Ry / Rx| with the quadrant fixed from the signs of Rx and Ry.
Worked example
Three forces act at a point: 5 N along +x, 4 N along +y, and 8 N along −x. Find the resultant.
Step 1 — components. The forces already lie along the axes, so no trigonometry is needed: (5, 0), (0, 4), (−8, 0).
Step 2 — sum x. Rx = 5 + 0 − 8 = −3 N.
Step 3 — sum y. Ry = 0 + 4 + 0 = 4 N.
Step 4 — rebuild. R = √(9 + 16) = 5 N. The acute angle is tan−1(4/3) = 53.1°, and since Rx < 0 with Ry > 0 the resultant lies in the second quadrant: θR = 180° − 53.1° = 126.9°.
Had you trusted the calculator’s tan−1(4 / −3) directly, it would have returned −53.1° — a resultant pointing down and to the right, when the true one points up and to the left. Same magnitude, opposite quadrant, no marks.
5. Subtraction and relative vectors
Subtraction is addition of a reversed vector: A − B = A + (−B), where −B has the same magnitude as B and points the opposite way. Running the cosine formula on that construction flips one sign:
|A − B| = √A² + B² − 2AB cos θ
Minus before the 2AB term — the only difference from the addition formula
The physically important use of this is relative motion. The velocity of A as seen by B is
vA/B = vA − vB
The subtraction is doing something specific, and it is worth seeing rather than memorising. To watch the world from B’s point of view, you have to bring B to rest — and you do that by subtracting B’s velocity from everything, including B itself. What remains is what B sees. This single idea is what makes rain-and-umbrella problems, river crossings and aircraft drift solvable, and all three are the same problem wearing different clothes.
Key insight
Resolution is not a computational trick applied to a vector. It is the claim that a vector is fully replaced by its components along any perpendicular pair you care to choose — and choosing that pair well is where most of the work in a mechanics problem gets done or wasted.
Common misconceptions
1. Magnitudes add: |A + B| = |A| + |B|
You met this in Foundation 2 as a statement about vectors. You can now see why it fails: the cosine formula reaches A + B only when cos θ = 1, that is when the vectors are parallel. With A = 3 and B = 4, a resultant of 8 is not merely unlikely — it is geometrically impossible.
2. A component can exceed the vector
For perpendicular axes it cannot: |Ax| = |A cos θ| ≤ A, since cosine never exceeds one. If a component comes out larger than the vector, an angle has been measured from the wrong axis.
3. The x-component always uses cosine
Cosine belongs to the axis adjacent to the angle you were given. If θ is measured from the y-axis, then Ax = A sin θ and Ay = A cos θ. Draw the triangle before writing anything; this single habit prevents most component errors.
4. Angles given as bearings can be used directly
A bearing of 30° east of north is not 30° from the x-axis — it is 60°. Navigation angles are measured clockwise from north; mathematical angles are measured anticlockwise from the x-axis. Convert first, resolve second.
Reflection
- If the components of a projectile’s motion never interact, in what sense is the trajectory a single thing rather than two?
- You are free to choose any perpendicular pair of axes. What, in a given problem, tells you which choice is the good one — before you have done the algebra?
- Relative velocity is a subtraction. What does it mean that there is no such thing as velocity without an implied observer?
Key takeaways
- A resultant is bounded: |A − B| ≤ R ≤ A + B. Use it to check every answer.
- The plus sign in R = √(A² + B² + 2AB cos θ) comes from the interior angle being 180° − θ.
- When A = B, the resultant is 2A cos(θ/2) along the bisector.
- Resolution replaces a vector with components along any perpendicular pair you choose; the choice is yours and it matters.
- Perpendicular components evolve independently because the equations are linear — not because forces are independent by nature.
- Subtraction gives relative quantities: vA/B = vA − vB is the world seen from B.
Prerequisite: Foundation 3 · Reading: 18 min · Practice: 25 min · Difficulty: Beginner