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Cross Products in the Light of Linear Transformations

The lecture connects the cross product’s determinant formula to its geometric meaning using linear transformations and duality.

3Blue1Brown⏱ 13 minOpen on YouTube ↗

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What the lecture covers

The lecture revisits the 3D cross product: its length is the area of the parallelogram spanned by two vectors, and its direction is perpendicular to both, with orientation set by the right-hand rule. To explain why the familiar determinant computation produces a vector with these properties, the speaker uses duality: every linear transformation from a vector space to the number line can be represented as a dot product with a corresponding vector.

Fix vectors v and w, and define a function of a third vector x by taking the determinant of the matrix with columns x, v, and w. This function is linear, and geometrically it gives the signed volume of the parallelepiped formed by the three vectors. By duality, the function equals the dot product of x with a particular vector p. Expanding the determinant identifies the coordinates of p with the coefficients produced by the usual cross-product calculation.

Geometrically, the signed volume is the area of the parallelogram formed by v and w multiplied by the component of x perpendicular to that parallelogram. Therefore p must be perpendicular to v and w and have length equal to their parallelogram’s area; its orientation accounts for the sign. The computational and geometric descriptions are thus two ways of finding the same dual vector, namely the cross product.

Key ideas

The cross product has magnitude equal to the area spanned by its inputs and points perpendicular to both according to the right-hand rule.
Duality represents any linear map to the number line as a dot product with a vector in the original space.
With v and w fixed, the determinant with a variable first column defines a function from 3D space to the number line.
The determinant function is linear, so its dual vector can be found either by computation or geometry.
Expanding the determinant shows that the dual vector’s coordinates are the combinations of v and w used in the cross-product formula.
The signed volume is the area spanned by v and w multiplied by the perpendicular component of the third vector.
The dual vector is perpendicular to v and w, has length equal to their parallelogram’s area, and has the orientation required for the sign.

Sample questions

For vectors v and w, the cross product has magnitude equal to the area of their parallelogram and points perpendicular to both, with its orientation set by the right-hand rule. If v and w are exchanged, which change follows from these properties?

AThe magnitude stays the same, but the direction reverses.
BThe magnitude reverses sign, but the direction stays the same.
CBoth the magnitude and direction stay the same.
DThe magnitude stays the same, and the direction becomes parallel to both vectors.

Show answer

Correct answer: A. Exchanging the vectors preserves the parallelogram’s area and the perpendicular line, but reverses the order used by the right-hand rule, so the oriented direction flips.

A linear map from [?] to the real numbers is represented by a one-row matrix with entries a₁, …, aₙ. Which vector lets you express the map’s value at any input v as a dot product?

ANo vector can represent a linear map to the real numbers using a dot product
BThe vector (a₁, …, aₙ), formed by turning the row of coefficients into a column
CA vector whose entries are the reciprocals of a₁, …, aₙ
DThe vector v itself, regardless of the map’s coefficients

Show answer

Correct answer: B. The coefficients of the one-row matrix, arranged as a vector, give a dot-product representation of the linear map.

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