References: Cross products as signed area

Source material

Source curriculum (structural mirror, cited as further study):
• 3Blue1Brown, Essence of Linear Algebra, Chapter 10: "Cross products"
  Creator: Grant Sanderson
  Lesson page: https://www.3blue1brown.com/lessons/cross-products
  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 (3Blue1Brown) by Grant Sanderson. The video this lesson mirrors. Watching the parallelogram’s signed area change as you rotate one vector, then watching it pass through zero as the vectors line up and flip sign as the orientation reverses, makes “signed area” tangible. The determinant connection is also drawn out explicitly. About nine minutes.

Going deeper

• Essence of Linear Algebra (full series) by 3Blue1Brown. The series this track follows. The determinant chapter is the one this lesson ties back to (the 2D cross product is that determinant); the next (Cross products in the light of linear transformations) lifts the cross product into 3D through a duality argument.

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

Adjacent topics

Where this sits in the track.

• The determinant (earlier lesson). The 2D cross product is the determinant of the matrix whose columns are the two vectors. The collinear-collapse case here is the same zero-determinant collapse, which is also the dependent-columns case from the spans lesson and the rank-deficiency case from the inverses lesson.

• Cross products in the light of linear transformations (next lesson). The 2D cross product is a number; the 3D cross product is a vector. The next lesson derives the 3D version through the same duality argument from the dot-product lesson, where a transformation that outputs a number turns out to correspond to a unique vector.