Cross products as signed area
What you’ll learn
Section titled “What you’ll learn”This is lesson 10 of Track 4 (Visual Math: Linear Algebra), and it opens Phase 3 (advanced perspectives). The previous lesson gave you the dot product, which measures how much two vectors align; this lesson gives you its partner, the cross product, which measures how much they spread apart. By the end you will be able to compute the 2D cross product and read its sign as orientation: positive when the second vector is counterclockwise from the first, negative when clockwise, zero when they are collinear. You will also recognize a pleasant surprise: the 2D cross product is exactly the determinant of the matrix the two vectors form, tying this lesson back to several earlier ones. 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 10 of 15, the first lesson of Phase 3 (advanced perspectives). It pairs with Dot products and projection (the two fundamental products of vectors) and ties back to The determinant, since the 2D cross product is that determinant seen from a new angle. The next lesson, Cross products via linear transformations, lifts the cross product into 3D, where the answer becomes a vector, and derives it with the duality argument from the dot-product lesson.
Before you start
Section titled “Before you start”Prerequisites: Dot products and projection (its partner operation and the duality idea), and The determinant (the 2D cross product is a determinant). You should be comfortable computing a 2x2 determinant as ad - bc, because the cross-product formula is identical. The practice is pen and paper.
About the math
Section titled “About the math”The arithmetic is one short formula, a·d - b·c, the same computation as a 2x2 determinant. You will compute a few cross products, read each result’s sign as a turn direction, and confirm anti-commutativity by swapping inputs. The depth is conceptual: seeing that this number is a signed area, and that it is the determinant you already know wearing a different name.
By the end, you’ll be able to
Section titled “By the end, you’ll be able to”- Compute the 2D cross product a·d - b·c of two vectors
- Interpret its magnitude as the area of the spanned parallelogram and its sign as orientation
- Recognize the 2D cross product as the determinant of the matrix whose columns are the two vectors
- Explain anti-commutativity (v × w = -(w × v)) and contrast it with the symmetric dot product
- Anticipate that the 3D cross product returns a vector rather than a number
Time and difficulty
Section titled “Time and difficulty”- Read time: about 9 minutes
- Practice time: about 15 minutes (a compute-and-read-orientation exercise, an anti-commutativity and determinant drill, and flashcards)
- Difficulty: standard (a Phase 3 lesson; the arithmetic is a single determinant-style formula, the depth is geometric)