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The cross product is bilinear, alternating, and orthogonal to both factors

Statement

For all $u, v, w \in R^{3}$ , the cross product is bilinear and alternating, $⟨ u \times v, w ⟩ = \det [u v w]$ , and $u \times v$ is orthogonal to both $u$ and $v$ .

Facts & Assumptions

Given: Vectors $u, v, w \in R^{3}$ and a scalar $c \in R$ .

[L1]

The cross product is given by its three coordinate differences of products (The cross product in $R^{3}$ ), and the Euclidean inner product is the coordinate dot product (The Euclidean inner product $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ ).

[L2]

The determinant of a real $3 \times 3$ matrix is its signed permutation sum (For $n \geq 1$ , the determinant over a commutative ring by the Leibniz formula, and $∣ \det A ∣$ for a real matrix).

Proof

technique · direct

1.1

Substitution in [L1] gives $(u + u^{'}) \times v = u \times v + u^{'} \times v$ , $(c u) \times v = c (u \times v)$ , and the corresponding two identities in the second argument.

L1algebra

1.2

The same coordinate formula gives $u \times u = 0$ and $v \times u = - (u \times v)$ , so the product is alternating.

L1algebra

1.3

Expanding the dot product in [L1] yields $⟨ u \times v, w ⟩ = (u_{y} v_{z} - u_{z} v_{y}) w_{x} + (u_{z} v_{x} - u_{x} v_{z}) w_{y} + (u_{x} v_{y} - u_{y} v_{x}) w_{z}$ , which is the determinant $\det [u v w]$ by [L2].

L1L2algebra

2.1

Taking $w = u$ or $w = v$ makes the determinant have two equal columns and hence zero; therefore $u \times v$ is orthogonal to both factors. The calculation includes zero and parallel vectors.

step 1.3L2algebra∎

Depends on

Used by

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

M. E. Taylor, Introduction to Analysis in Several Variables, Section 3.2 (standard reference, not scraped)
University of Toronto MAT237 notes, Section 5.3 (standard reference, not scraped)