Multiplying vectors in 2-space

Mul2[A_, B_] :=  {A[[1]] B[[1]] + A[[2, 1]] B[[2, 1]] + A[[2, 2]] B[[2, 2]] ... 1;}, A[[1]] B[[3]] + A[[3]] B[[1]] + A[[2, 1]] B[[2, 2]] - A[[2, 2]] B[[2, 1]] }

Mul2[a2, b2]//MatrixForm

( {{a0 b0 - ai bi + ax bx + ay by}, {{ax b0 - ay bi + a0 bx + ai by, ay b0 + ax bi - ai bx + a0 by}}, {ai b0 + a0 bi - ay bx + ax by}} )

Mul2[avec, bvec] //MatrixForm(* multiplying two vectors to get scalar and bivector components *)

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

Mul2[{0, {ax, 0}, 0}, {0, {bx, 0}, 0}]//MatrixForm

( {{ax bx}, {{0, 0}}, {0}} )

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