Deriving the 3D cross product from duality
What you’ll learn
Section titled “What you’ll learn”This is lesson 11 of Track 4 (Visual Math: Linear Algebra). The 3D cross product is usually handed over as a formula to memorize, three components each a different criss-cross of products. This lesson does the opposite: it derives that formula from scratch, and by the end you will be able to reconstruct it from a single definition rather than memorize it. The engine is the duality idea from the dot-product lesson, lifted one dimension and applied to the signed-volume function. You will see the cross product defined as the unique vector whose dot with any p gives the volume of the box p makes with v and w, and you will watch its three famous properties (perpendicular, length equals area, right-hand direction) fall out as automatic consequences. The source curriculum is 3Blue1Brown’s Essence of Linear Algebra by Grant Sanderson, freely available at 3blue1brown.com.
Where this fits
Section titled “Where this fits”This is lesson 11 of 15, the second lesson of Phase 3 (advanced perspectives). It is the second act of duality: Dot products and projection showed that a linear map to numbers is a vector in disguise, and this lesson applies that exact idea to a volume function to produce the 3D cross product. It builds on Cross products as signed area (the 2D version) and The determinant (signed volume). The next lesson, Cramer’s rule, reuses the same determinant-as-volume tools to solve linear systems.
Before you start
Section titled “Before you start”Prerequisites: Cross products as signed area (the 2D cross product and its determinant link) and, importantly, Dot products and projection (the duality argument this lesson lifts into 3D). It also helps to recall from The determinant that a 3x3 determinant is a signed volume. This is the most conceptually demanding lesson in the track; the prerequisites are doing real work. The practice is pen and paper.
About the math
Section titled “About the math”This lesson is more derivation than computation. You will set up a volume function, invoke duality, and read the cross-product components off a determinant expansion. The practice has you compute a few 3D cross products with the formula and verify perpendicularity by dotting, plus work the cyclic basis pattern. The arithmetic per problem is light (products and differences); the demand is conceptual, in following why the definition forces the formula and the properties.
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- State the three target properties of the 3D cross product (perpendicular, length equals area, right-hand direction)
- Set up the signed-volume function f(p) = det([p | v | w]) and recognize it as linear in p
- Apply duality to define v × w as the unique vector u with u·p equal to that signed volume
- Read off the cross-product formula as the cofactor expansion of the determinant, without memorizing it
- Explain why perpendicularity and the length-equals-area property are forced consequences of the definition
Time and difficulty
Section titled “Time and difficulty”- Read time: about 12 minutes
- Practice time: about 15 minutes (a compute-and-verify-perpendicularity exercise, a basis cross-product and anti-commutativity drill, and flashcards)
- Difficulty: standard, on the harder end (the most conceptually demanding lesson in the track; the arithmetic stays light but the duality derivation asks for real attention)