Why Cross-Product Coordinates Match Their Geometry

要約

30 秒でわかる

Grant Sanderson relates the cross product to the linear map u ↦ det[u, v, w]. Expanding the determinant gives its coordinates, while signed volume explains its perpendicular direction and magnitude.

1GeometryThe cross product encodes perpendicular direction and parallelogram area.
2DualityA linear functional corresponds to a vector through the dot product.
3Signed volumeFixing v and w makes det[u, v, w] linear in u.
4CoordinatesIts coefficients give the cross-product coordinates.
5Why they agreeArea times signed height yields the same dot product.

ポイント

The cross product encodes perpendicular direction and parallelogram area.
A linear functional corresponds to a vector through the dot product.
Fixing v and w makes det[u, v, w] linear in u.
Its coefficients give the cross-product coordinates.
Area times signed height yields the same dot product.

トピックMathematics Education

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Cross products and duality

From the determinant mnemonic to signed volume.

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.