Module 1: Mathematical Foundations · Theory · assumes Foundation 5
The second way to multiply two vectors — and the one that gives rotation something to point at.
Mahavakya
The cross product points somewhere no vector needs to point. We agree on which way — and physics works because we all agree.
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Learning objectives
After this reading you should be able to:
- State the cross product geometrically and connect its magnitude to the area of a parallelogram.
- Apply the right-hand rule correctly, and explain why a rule is needed at all.
- Compute a cross product by determinant expansion, and check the answer with a dot product.
- Use anti-commutativity and non-associativity without treating them as things to memorise.
- Compute torque as force times perpendicular distance, and say why torque depends on the pivot you choose.
Notation. Vectors in bold: A, B, C. The cross uses a multiplication sign: A × B. Unit vectors carry a hat: î, ĵ, k̂, and n̂ is the unit vector perpendicular to a plane. The zero vector is written 0, which is not the same object as the number 0.
1. Rotation, and area
Foundation 5 took two vectors and asked how much they agree. The answer was a scalar, and it described things like work — quantities with no direction of their own. But mechanics also deals with turning: tightening a bolt, a planet sweeping round the sun, a spinning wheel. For these, the answer needs to be a vector, and it needs to know about the part of one vector that is perpendicular to the other. That is the second product.
|A × B| = AB sin θ
A × B = (AB sin θ) n̂
θ is the angle between them, and n̂ is perpendicular to the plane they span
Where the dot product peaked when the vectors were aligned, this one vanishes there. Parallel or anti-parallel, sin θ = 0 and the result is the zero vector. Perpendicular, sin θ = 1 and the magnitude reaches its largest possible value, AB. The two products are measuring opposite things about the same pair.
The magnitude has a clean geometric meaning: it is the area of the parallelogram that A and B span. Base A, height B sin θ, area AB sin θ. This is worth holding on to, because it explains the vanishing case without any algebra — two parallel vectors span no parallelogram at all, only a line, and a line has no area.
Two vectors, one plane, two possible perpendiculars. Something has to choose.
2. The right-hand rule
Look again at that diagram. A and B lie in a plane, and there are exactly two directions perpendicular to it — one up, one down. Nothing about A or B favours either. The formula AB sin θ fixes how long the answer is; it says nothing at all about which way it points.
So we choose. The choice is called the right-hand rule:
- Point the fingers of your right hand along A.
- Curl them toward B, through the smaller of the two angles between them — which may be obtuse, up to 180°, but never the reflex angle.
- Your thumb points along A × B.
Sweep the other way, from B to A, and your thumb reverses. This gives the property that most distinguishes this product from the last one:
A × B = −(B × A)
Anti-commutative — order changes the answer
Note what kind of statement the right-hand rule is. It is not a discovery about nature and it cannot be derived. If every physicist had settled on the left hand instead, every cross product in every textbook would point the opposite way, and every prediction physics makes would be unchanged — because the reversals would occur in matched pairs and cancel. What matters is that everyone uses the same hand.
3. The component form
Applying the geometric definition to the basis vectors themselves gives everything needed for computation. Each is perpendicular to the other two, so sin θ = 1 between different ones, and each is parallel to itself, so sin θ = 0 there:
î × î = ĵ × ĵ = k̂ × k̂ = 0
î × ĵ = k̂, ĵ × k̂ = î, k̂ × î = ĵ
Going forwards round î → ĵ → k̂ → î gives a plus sign; going backwards gives a minus
Expanding two general vectors with those rules produces a result that can be written as a determinant — not because determinants have anything to do with rotation, but because the pattern of plus and minus signs happens to match:
| î | ĵ | k̂ |
| Ax | Ay | Az |
| Bx | By | Bz |
= (AyBz − AzBy) î − (AxBz − AzBx) ĵ + (AxBy − AyBx) k̂
The minus sign on the middle term is not optional and is the single most common source of lost marks in this topic. It comes from the alternating signs of determinant expansion, not from anything physical.
Worked example
Given A = 2î + ĵ − 3k̂ and B = 4ĵ + 2k̂, find A × B.
î component: (1)(2) − (−3)(4) = 2 + 12 = 14.
ĵ component: −[(2)(2) − (−3)(0)] = −[4 − 0] = −4.
k̂ component: (2)(4) − (1)(0) = 8.
So A × B = 14î − 4ĵ + 8k̂.
Check it. The answer must be perpendicular to both inputs, and Foundation 5 gives you a one-line test. A · (A × B) = 28 − 4 − 24 = 0, and B · (A × B) = 0 − 16 + 16 = 0. Both zero, so the answer survives. Do this every time — it catches sign errors in seconds.
4. Properties
- Anti-commutative. A × B = −(B × A). Order is not a formality here.
- Distributive. A × (B + C) = A × B + A × C, which is what makes the component expansion legal.
- Self-product vanishes. A × A = 0, since a vector is parallel to itself. Compare Foundation 5, where A · A = A².
It is also not associative, and this is worth seeing rather than being told. Take the simplest possible case:
(î × î) × ĵ = 0 × ĵ = 0
î × (î × ĵ) = î × k̂ = −ĵ
Same three vectors, same order, different brackets — and the answers are not merely different in sign but different in kind, one being the zero vector and the other a unit vector. Brackets are load-bearing.
One more relationship deserves attention. Squaring both products and adding gives:
|A × B|² + (A · B)² = A²B²
Because A²B²(sin²θ + cos²θ) = A²B²
Key insight
The dot and cross products are not two unrelated tools that happen to involve two vectors. Between them they account for everything: one takes the parallel part, the other the perpendicular part, and the identity above says nothing is left over. Give me both products and I can reconstruct the magnitudes and the angle. Give me only one and I cannot.
5. Where it appears in mechanics
Torque
τ = r × F, where r runs from the pivot to the point where the force acts. The magnitude is rF sin θ.
In practice you rarely compute that with trigonometry. Group it as F × (r sin θ), and the bracket is the perpendicular distance — the shortest distance from the pivot to the line along which the force acts, extended in both directions if necessary. Torque is then simply force times perpendicular distance, which is how seesaw, ladder and equilibrium problems are actually solved.
Boundary: torque is defined relative to a chosen pivot. Move the pivot and r changes, so τ changes. There is no such thing as “the” torque on a body — only the torque about a stated point. Always state it.
Angular momentum
L = r × p = r × mv, the rotational counterpart of linear momentum. It carries the same origin-dependence as torque, for the same reason.
Velocity of a rotating point
v = ω × r. A point on a spinning body moves perpendicular to both the axis and its own position vector — which is exactly what a cross product produces, without anyone having to impose it.
Contemplation
Ask where a torque vector actually points and the honest answer is unsettling: nowhere in particular. The turning happens in a plane. The arrow sticking out of that plane is not describing any motion — nothing travels along it, nothing is pushed that way. It is a device for naming the plane and the sense of turning within it, using an arrow because arrows are what our mathematics knows how to handle.
There is a test that exposes the difference. Reverse all three coordinate axes at once, so that every point in space maps to its opposite. Genuine directional quantities — force, velocity, displacement — reverse with it, as you would expect. Torque and angular momentum do not. They come through unchanged, because each is built from two vectors that both flipped, and the two sign changes cancel. Quantities that behave this way are called axial vectors, or pseudovectors, and they are a different kind of thing from a force.
Foundation 3 said the coordinate frame belongs to our method rather than to nature. This is that claim made sharper and less comfortable: the direction of a physical quantity can belong to our method too. Force points where it points whatever we decide. Torque points where it does because we agreed on a hand.
One boundary worth stating. Handedness is a convention throughout classical mechanics, and everything above holds without qualification for this course. At a much deeper level nature is not entirely indifferent — the weak nuclear interaction distinguishes left from right in a way no convention can absorb. That belongs to a physics far beyond this module, and it changes nothing here. It is mentioned only because “purely a convention” is itself a claim, and claims have boundaries.
Common misconceptions
1. Order does not matter
It reverses the answer. A × B and B × A have the same magnitude and opposite directions. Writing F × r when you meant r × F gives a torque pointing the wrong way, which in a rotational problem means the body turns the wrong way.
2. There is a cross product in two dimensions
There cannot be. The result must be perpendicular to the plane of the inputs, and a genuinely two-dimensional space has no direction perpendicular to itself. What you meet in problems is two vectors lying in the xy-plane of a three-dimensional space, whose cross product points along k̂ — it leaves the plane. The cross product is intrinsically three-dimensional in a way the dot product is not.
3. A × A should be A²
That is the dot product’s behaviour. Here the angle is zero, sin 0° = 0, and the result is the zero vector — not the number zero. The distinction matters when the answer feeds into another vector equation.
4. Sine and cosine are interchangeable if you are careful
They select opposite things. Cosine keeps the aligned part and is maximal for parallel vectors; sine keeps the perpendicular part and is maximal at right angles. A force applied along a wrench handle produces no torque at all, and using cos θ would predict the maximum.
Reflection
- If a left-handed convention would work equally well, is there anything left that is physically true about the direction of a torque vector?
- Two vectors in a plane produce a result outside it. What does that suggest about why rotation is harder to visualise in three dimensions than in two?
- The dot product discards direction to keep a magnitude; the cross product discards nothing but adds a direction that was never in the inputs. Which of the two is doing the stranger thing?
Key takeaways
- A × B = (AB sin θ) n̂, a vector perpendicular to both inputs.
- Its magnitude is the area of the parallelogram they span — zero when they are parallel.
- The direction comes from the right-hand rule, which is an agreement, not a derivation.
- Anti-commutative, distributive, and not associative; A × A = 0.
- Check every computed cross product by dotting it with both inputs; both must give zero.
- Torque is force times perpendicular distance, and is always about a stated pivot.
- Torque and angular momentum are axial vectors — they do not reverse under inversion, and a force does.
Prerequisite: Foundation 5 · Reading: 18 min · Practice: 25 min · Difficulty: Intermediate