Definition

$v \times w = ∣∣ v ∣∣∣∣ w ∣∣ sin (θ) n = (v_{2} w_{2} v_{3} w_{3}, - v_{1} w_{1} v_{3} w_{3}, v_{1} w_{1} v_{2} w_{2}) = [(v_{2} w_{3} - v_{3} w_{2}), (v_{3} w_{1} - v_{1} w_{3}), (v_{1} w_{2} - v_{2} w_{1})] = det i j k v_{1} v_{2} v_{3} w_{1} w_{2} w_{3}$ where the $v, w \in R^{3}$ , the $θ$ is the angle between $v$ and $w$ , and the $n$ is a unit vector perpendicular to the both $v$ and $w$ .

Facts

$u \times v = - (v \times u)$

$u \times (v + w) = (u \times v) + (u \times w)$ $u \times (v - w) = (u \times v) - (u \times w)$

$(u + v) \times w = (u \times w) + (v \times w)$

$k (u \times v) = (k u) \times v = u \times (k v)$

$u \times 0 = 0 \times u = 0$

$u \times u = 0$

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