Cross Product Tools

Let

A = a₁i + a₂j + a₃k

B = b₁i + b₂j + b₃k

Cyclic Order

i → j → k → i

i × j = k j × k = i k × i = j

i × k = −j k × j = −i j × i = −k

Coordinate Checks

(1, 0, 0) × (0, 1, 0) (0, 0, 1)
(0, 1, 0) × (0, 0, 1) (1, 0, 0)
(0, 0, 1) × (1, 0, 0) (0, 1, 0)
(1, 0, 0) × (0, 0, 1) (0, −1, 0)
(0, 0, 1) × (0, 1, 0) (−1, 0, 0)
(0, 1, 0) × (1, 0, 0) (0, 0, −1)

Scalar Products

× b₁ b₂ b₃
a₁ a₁b₁ a₁b₂ a₁b₃
a₂ a₂b₁ a₂b₂ a₂b₃
a₃ a₃b₁ a₃b₂ a₃b₃

Basis-Vector Products

× i j k
i 0 k −j
j −k 0 i
k j −i 0

Cross Product Expansion

The cross product distributes over vector addition, and the scalar factors may be extracted.

(a₁i + a₂j + a₃k)

×

(b₁i + b₂j + b₃k)

a₁b₁(i × i) a₁b₂(i × j) a₁b₃(i × k)
a₂b₁(j × i) a₂b₂(j × j) a₂b₃(j × k)
a₃b₁(k × i) a₃b₂(k × j) a₃b₃(k × k)

Substituting the basis-vector products gives

a₁b₁(0) a₁b₂k −a₁b₃j
−a₂b₁k a₂b₂(0) a₂b₃i
a₃b₁j −a₃b₂i a₃b₃(0)

Collecting the i, j and k components gives

A × B

= (a₂b₃ − a₃b₂)i
+ (a₃b₁ − a₁b₃)j
+ (a₁b₂ − a₂b₁)k

Coordinate Form

A × B

= (a₂b₃ − a₃b₂,
a₃b₁ − a₁b₃,
a₁b₂ − a₂b₁)

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