Summary: Deriving the 3D cross product

The 3D cross product has a formula that looks like pure memorization: three components, each a different criss-cross of products. This lesson refuses to memorize it and derives it instead. The whole thing reduces to one line: the 3D cross product is the unique vector whose dot with any p gives the signed volume of the box p makes with v and w, and that definition hands you the formula and all three geometric properties for free. This is the scan-it-in-five-minutes version.

Core ideas

• The target: v × w should be a vector (not a number, unlike the 2D case) that is perpendicular to both inputs, has length equal to the area of the parallelogram they span, and points by the right-hand rule.
• The derivation starts with a function: fix v and w, and let f(p) = det([p | v | w]), the signed volume of the parallelepiped spanned by p, v, and w. This is the 3D version of “determinant = signed area.” It is a linear function from a 3D vector p to a single number.
• Duality, the idea from the dot-product lesson lifted one dimension, says every linear function from 3D to a number is a dot product with one unique vector. So there must be a vector u with u · p = det([p | v | w]) for every p. That u is the definition of v × w.
• The formula is then read off, not memorized: expand det([p | v | w]) along the p column (a cofactor expansion) and match coefficients to u · p, giving [v_y·w_z - v_z·w_y, v_z·w_x - v_x·w_z, v_x·w_y - v_y·w_x]. The criss-cross pattern is just the cofactor terms.
• The three properties are automatic. Perpendicular: u · v is the volume of a box with a repeated edge ([v | v | w]), which is flat, so u · v = 0. Length = area: with p a unit vector along u, the box is a height-1 prism on the parallelogram base, so |u| equals that base area. Right-hand direction: from the signed-volume convention.
• Worked anchors: i × j = [0,0,1] = k (and j × i = -k, anti-commutative); [1,2,3] × [4,5,6] = [-3,6,-3], which dots to zero against both inputs; [2,0,0] × [0,3,0] = [0,0,6], length 6 equal to the 2-by-3 rectangle’s area. The basis pattern is cyclic: i × j = k, j × k = i, k × i = j.
• Honest about AI: as with the 2D cross product, this is less central than the dot product (physics-informed networks, 3D geometric deep learning, robotics, quaternion rotations). The deeper takeaway is the method: duality manufacturing a vector out of a volume function, a pattern that recurs throughout the mathematics under machine learning.

What changes for you

Before this lesson, the 3D cross-product formula was probably something to look up and hope you copied the signs correctly. Now you have a definition you can reconstruct it from: the vector whose dot with any p gives a volume. More importantly, you have seen duality work twice, once to explain the dot product and once to build the cross product from a volume function, so the abstract idea “a linear map to numbers is secretly a vector” now has two concrete payoffs behind it. The next lesson reuses the same determinant-as-volume picture to crack open how linear systems are solved, through Cramer’s rule.