Why Cross-Product Coordinates Match Their Geometry

In this Essence of Linear Algebra chapter, Grant Sanderson explains why the usual computation of a three-dimensional cross product agrees with its geometric interpretation. The video description corrects the third line of the displayed matrix at 1:44 to v₁w₂ − w₁v₂.

The familiar mnemonic places basis-vector symbols in the first column of a determinant-like array, with v and w in the other columns. Expanding it gives three coefficients, interpreted as the coordinates of v × w.

Geometrically, v × w is perpendicular to both vectors, has length equal to the area of their parallelogram, and follows the right-hand rule. When the vectors are parallel, the cross product is zero, so it has no unique direction.

The bridge is duality in Euclidean space: a linear map from vectors to real numbers can be represented by a dot product with a unique vector.

Fix v and w, and define a function of u by det[u, v, w]. Its value is the signed volume of the parallelepiped spanned by the three vectors. Because the determinant is linear in u, the function has a corresponding dual vector.

Expanding the determinant in the coordinates of u gives coefficients that are exactly the coordinates produced by the cross-product mnemonic. Thus the associated vector p satisfies u · p = det[u, v, w] for every u.

The same signed volume equals the area spanned by v and w times the signed component of u perpendicular to their plane. A dot product produces precisely this quantity when p has that perpendicular direction and area as its length.

Both the coordinate calculation and the geometric construction represent the same linear function. Their corresponding vector is therefore the same cross product.