References: Deriving the 3D cross product

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 11: "Cross products in the light of linear transformations"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/cross-products-extended
  Series index: https://www.3blue1brown.com/?topic=linear-algebra
  License: copyright Grant Sanderson; videos published on his site and YouTube
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Watch this next

• Cross products in the light of linear transformations (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors, and one of the most elegant in the series. Seeing the volume function get built and then collapse, via duality, into a single perpendicular vector is far more convincing animated than on the page. If any lesson in this track rewards watching the original, it is this one. About fourteen minutes.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The dot-product chapter supplied the duality insight that drives this derivation, and the determinant chapter supplied the signed-volume picture; the next (Cramer’s rule, explained geometrically) reuses the same determinant-as-volume tools to solve linear systems.

• Khan Academy: Linear algebra for a slower, exercise-driven treatment of the cross product and its computation, with practice problems and immediate feedback.

Adjacent topics

Where this sits in the track.

• Dot products and duality (earlier lesson). This lesson is duality’s second act. The dot-product lesson showed that a linear map to numbers is a vector in disguise; here that exact idea, applied to a volume function, produces the cross product. The two lessons are one idea used twice.

• Cramer’s rule, explained geometrically (next lesson). The determinant-as-volume picture that defined the cross product here is the same tool that cracks open how linear systems are solved. Cramer’s rule expresses each unknown as a ratio of determinants, which are ratios of volumes, returning to the solve-for-the-input question from the inverses lesson with sharper geometric tools.