The Outer Product $a \land b$ is the oriented parallelogram with sides $a$ and $b$ . It is the fundamental operation of Exterior Algebra.

Oriented means swapping the sides reverses the orientation:

$b \land a = - (a \land b)$

From antisymmetry, $a \land a = 0$ (a degenerate parallelogram has no area). The result of the outer product of two vectors is a Bivector (grade 2).

Axioms

Distributivity: $a \land (b + c) = (a \land b) + (a \land c)$
Scalar associativity: $α (a \land b) = (α a) \land b = a \land (α b)$
Antisymmetry: $b \land a = - (a \land b)$

Calculation

Expand in the standard basis and apply the axioms. All $e_{i} \land e_{i} = 0$ terms drop, and $e_{j} \land e_{i} = - e_{i} \land e_{j}$ for $i \neq = j$ , so the result is a linear combination of basis bivectors $e_{i} \land e_{j}$ for $i < j$ .

In $R^{2}$ with $a = a_{1} e_{1} + a_{2} e_{2}$ and $b = b_{1} e_{1} + b_{2} e_{2}$ :

$a \land b = (a_{1} b_{2} - a_{2} b_{1}) (e_{1} \land e_{2}) = ∣ a ∣∣ b ∣ sin θ (e_{1} \land e_{2})$

Every bivector in $R^{2}$ is a scalar multiple of $e_{1} \land e_{2}$ .

In $R^{3}$ with $a = a_{1} e_{1} + a_{2} e_{2} + a_{3} e_{3}$ and $b = b_{1} e_{1} + b_{2} e_{2} + b_{3} e_{3}$ :

$a \land b = a_{1} b_{1} a_{2} b_{2} (e_{1} \land e_{2}) + a_{2} b_{2} a_{3} b_{3} (e_{2} \land e_{3}) + a_{1} b_{1} a_{3} b_{3} (e_{1} \land e_{3})$

In terms of the Geometric Product

$a \land b = \frac{1}{2} (a b - b a)$

Notes

Explorer

Outer Product

Axioms

Calculation

In terms of the Geometric Product

Graph View

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