Dot and wedge products in 3 space

DotPr3[a_, b_] := 1/2 (Mul3[a, b] + Mul3[b, a])

Simplify[DotPr3[a3v, b3v]]//MatrixForm

( {{ax bx + ay by + az bz}, {{0, 0, 0}}, {{0, 0, 0}}, {0}} )

Simplify[DotPr3[a3, b3]]//MatrixForm

( {{a0 b0 - ai bi - aix bix - aiy biy - aiz biz + ax bx + ay by + az bz}, {{ax b0 - ... biz + ai bz}}, {ai b0 + a0 bi + ax bix + ay biy + az biz + aix bx + aiy by + aiz bz}} )

Wedge3[a_, b_] := 1/2 (Mul3[a, b] - Mul3[b, a])

Simplify[Wedge3[a3v, b3v]]//MatrixForm

( {{0}, {{0, 0, 0}}, {{-az by + ay bz, az bx - ax bz, -ay bx + ax by}}, {0}} )

Simplify[Wedge3[a3, b3]]//MatrixForm

( {{0}, {{az biy - ay biz + aiz by - aiy bz, -az bix + ax biz - aiz bx + aix bz, ay ... + ay bz, -aiz bix + aix biz + az bx - ax bz, aiy bix - aix biy - ay bx + ax by}}, {0}} )

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