notation ... dot product of two vectors: A . B cross product of three vectors: A x B x C multiplication of vector A by scalar Z: ZA complex conjugate: ~A define some arbitrary vectors ... A = [ (-0.58095, +0.54826) (-0.43443, -0.14279) (+0.34092, +0.08455) (+0.43339, +0.01640) (+0.69312, +0.65326) (-0.10353, +0.21420) (+0.64615, +0.50118) (-0.28836, -0.11950) ] B = [ (+0.02085, +0.40570) (+0.06198, -0.56892) (-0.09484, +0.48290) (+0.38199, -0.43140) (+0.39251, -0.44612) (+0.11080, -0.26022) (-0.12322, +0.68923) (-0.47051, -0.23121) ] C = [ (-0.59329, -0.45567) (-0.64778, -0.04562) (+0.24376, -0.60662) (-0.15430, -0.46844) (+0.68713, +0.51889) (+0.24171, +0.12279) (-0.34298, +0.13026) (-0.31959, -0.32133) ] D = [ (-0.56570, +0.18558) (-0.48822, +0.62841) (-0.64035, +0.32912) (-0.04830, -0.38844) (+0.10008, +0.14640) (+0.27720, +0.23056) (-0.45689, +0.35209) (+0.45488, +0.35433) ] E = [ (-0.25333, +0.04557) (-0.49342, +0.40968) (+0.57754, +0.41213) (-0.68683, +0.01407) (-0.28617, +0.08389) (+0.00584, -0.26598) (+0.67561, +0.65024) (-0.07201, -0.27508) ] | A | = +1.69958 | B | = +1.50495 | C | = +1.67537 | D | = +1.57669 | E | = +1.60397 define some arbitrary scalars ... X = (-0.62131, +0.42269) Y = (+0.67734, -0.62851) Z = (-0.36428, -0.55342) ===== essential cross product identities ===== orthogonality ... should be zero: A . (A x B x C) = (-0.00000, +0.00000) B . (A x B x C) = (+0.00000, -0.00000) C . (A x B x C) = (+0.00000, +0.00000) (A x B x C) . A = (-0.00000, +0.00000) (A x B x C) . B = (+0.00000, -0.00000) (A x B x C) . C = (+0.00000, +0.00000) gram determinant ... should be equal: (A x B x C) . (A x B x C) = (+1.71422, +0.95357) gram determinant = (+1.71422, +0.95357) ===== other cross product identities ===== anti-commutativity ... should be equal: + (A x B x C) = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] + (B x C x A) = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] + (C x A x B) = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] - (A x C x B) = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] - (B x A x C) = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] - (C x B x A) = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] trivalued associativity ... should be equal: + ((A x B x C) x A x B) = [ (+2.13861, -0.01369) (+0.54209, -0.89478) (+0.83575, +0.41504) (+1.24944, -1.05630) (-0.04351, -0.46180) (+0.26327, +0.34546) (+0.01041, -1.54119) (+0.40952, -0.34957) ] - (A x (B x C x A) x B) = [ (+2.13861, -0.01369) (+0.54209, -0.89478) (+0.83575, +0.41504) (+1.24944, -1.05630) (-0.04351, -0.46180) (+0.26327, +0.34546) (+0.01041, -1.54119) (+0.40952, -0.34957) ] + (A x B x (C x A x B)) = [ (+2.13861, -0.01369) (+0.54209, -0.89478) (+0.83575, +0.41504) (+1.24944, -1.05630) (-0.04351, -0.46180) (+0.26327, +0.34546) (+0.01041, -1.54119) (+0.40952, -0.34957) ] cycling ... should be equal: + ((A x B x C) . D) = (+0.23481, -0.20927) - ((D x A x B) . C) = (+0.23481, -0.20927) + ((C x D x A) . B) = (+0.23481, -0.20927) - ((B x C x D) . A) = (+0.23481, -0.20927) distributivity ... should be equal: (A + B) x C x D = [ (+0.07053, -1.67730) (-1.30687, -0.03759) (-1.52810, -1.18517) (-1.69140, +0.80543) (-0.08083, +0.20546) (-3.95984, -1.07240) (-0.91775, +1.94064) (-0.64317, +3.53072) ] (A x C x D) + (B x C x D) = [ (+0.07053, -1.67730) (-1.30687, -0.03759) (-1.52810, -1.18517) (-1.69140, +0.80543) (-0.08083, +0.20546) (-3.95984, -1.07240) (-0.91775, +1.94064) (-0.64317, +3.53072) ] should be equal: A x (B + C) x D = [ (-0.55430, +0.64619) (+0.25937, -0.53279) (+0.29074, -1.87857) (-0.13073, -0.33414) (+0.80354, +0.26317) (-0.80445, -0.22929) (-0.44736, +1.03480) (-1.20109, -0.35022) ] (A x B x D) + (A x C x D) = [ (-0.55430, +0.64619) (+0.25937, -0.53279) (+0.29074, -1.87857) (-0.13073, -0.33414) (+0.80354, +0.26317) (-0.80445, -0.22929) (-0.44736, +1.03480) (-1.20109, -0.35022) ] should be equal: A x B x (C + D) = [ (-0.27001, +0.67058) (+0.30551, -1.25887) (+2.04402, -0.41604) (+1.43830, -1.20332) (-0.08588, +0.74850) (+1.31720, -0.49668) (-1.03505, -0.66172) (-0.98760, -1.84947) ] (A x B x C) + (A x B x D) = [ (-0.27001, +0.67058) (+0.30551, -1.25887) (+2.04402, -0.41604) (+1.43830, -1.20332) (-0.08588, +0.74850) (+1.31720, -0.49668) (-1.03505, -0.66172) (-0.98760, -1.84947) ] factoring out scalars ... should be equal: (XA x YB x ZC) = [ (-0.16929, -0.35536) (-0.22371, -0.08482) (+0.66171, -0.10958) (+0.04956, -0.35941) (-0.11188, +0.34655) (-0.07937, -0.42276) (-0.66403, +0.06442) (-0.13988, -0.11972) ] XYZ(A x B x C) = [ (-0.16929, -0.35536) (-0.22371, -0.08482) (+0.66171, -0.10958) (+0.04956, -0.35941) (-0.11188, +0.34655) (-0.07937, -0.42276) (-0.66403, +0.06442) (-0.13988, -0.11972) ] repeated factors ... should be zero: A x A x B = [ (-0.00000, +0.00000) (+0.00000, -0.00000) (+0.00000, +0.00000) (+0.00000, +0.00000) (+0.00000, +0.00000) (-0.00000, -0.00000) (+0.00000, -0.00000) (-0.00000, +0.00000) ] A x B x A = [ (+0.00000, +0.00000) (+0.00000, +0.00000) (+0.00000, +0.00000) (-0.00000, +0.00000) (+0.00000, +0.00000) (+0.00000, -0.00000) (+0.00000, -0.00000) (-0.00000, +0.00000) ] B x A x A = [ (-0.00000, +0.00000) (+0.00000, -0.00000) (+0.00000, +0.00000) (+0.00000, -0.00000) (-0.00000, +0.00000) (+0.00000, -0.00000) (-0.00000, -0.00000) (+0.00000, -0.00000) ] ortho_mag ... define some arbitrary orthogonal vectors: P = [ (+0.04872, +0.02997) (-0.23803, -0.25531) (+0.66900, +0.32912) (+0.63506, -0.63074) (+0.10155, -0.61482) (+0.43628, +0.67086) (-0.38384, -0.13108) (+0.16666, +0.35638) ] Q = [ (+0.58385, -0.65379) (+0.44624, +0.59141) (-0.44051, +0.02839) (-0.30932, -0.18792) (-0.10488, -0.43817) (-0.29167, +0.10192) (-0.42744, -0.46615) (+0.26397, +0.40010) ] R = [ (-0.43906, +0.19275) (-0.41735, +0.55927) (+0.23258, +0.75692) (-0.14994, -0.21370) (-0.27166, -0.05481) (-0.42677, -0.30220) (-0.17119, -0.28010) (+0.00076, -0.29456) ] | P | = +1.68230 | Q | = +1.60352 | R | = +1.39887 P x Q x R = [ (+0.82170, +0.16801) (-1.37054, +0.55615) (+0.53928, +0.04180) (-0.57578, -1.70935) (-1.35939, +0.20661) (+0.01492, -0.64372) (+0.07011, +0.77225) (+0.06740, -0.22646) ] should be zero: dot (P, ~Q) = (-0.00000, +0.00000) dot (Q, ~R) = (-0.00000, +0.00000) dot (R, ~P) = (+0.00000, +0.00000) should NOT be equal: | P x Q x R | = +3.06584 | P | | Q | | R | = +3.77362 slicing ... should be equal: A x B x C = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] by slicing = [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] ===== complex cross product identities ===== conjugation ... should be equal: (~A x ~B x ~C) = [ (-0.07506, +0.85231) (-0.39126, +0.34259) (+1.43097, -0.27914) (+0.37378, +0.69443) (-0.49096, -0.62092) (+0.15926, +0.92134) (-1.40142, +0.37288) (-0.19403, +0.35003) ] ~ ( A x B x C) = [ (-0.07506, +0.85231) (-0.39126, +0.34259) (+1.43097, -0.27914) (+0.37378, +0.69443) (-0.49096, -0.62092) (+0.15926, +0.92134) (-1.40142, +0.37288) (-0.19403, +0.35003) ] should be equal: (~A x B x ~C) = [ (-0.01375, +0.09667) (+1.00380, -0.52662) (-1.60638, -1.69057) (+0.28656, -1.28441) (-0.99431, +0.92419) (+0.20373, -0.52061) (+1.43611, +0.12282) (-0.32042, -0.19664) ] ~ ( A x ~B x C) = [ (-0.01375, +0.09667) (+1.00380, -0.52662) (-1.60638, -1.69057) (+0.28656, -1.28441) (-0.99431, +0.92419) (+0.20373, -0.52061) (+1.43611, +0.12282) (-0.32042, -0.19664) ] conjugative cycling ... should be equal: + (( A x ~B x C) . ~D) = (+0.06924, -1.43138) + (( C x ~D x A) . ~B) = (+0.06924, -1.43138) - ((~B x C x ~D) . A) = (+0.06924, -1.43138) - ((~D x A x ~B) . C) = (+0.06924, -1.43138) ~ + ((~A x B x ~C) . D) = (+0.06924, -1.43138) ~ + ((~C x D x ~A) . B) = (+0.06924, -1.43138) ~ - (( B x ~C x D) . ~A) = (+0.06924, -1.43138) ~ - (( D x ~A x B) . ~C) = (+0.06924, -1.43138) ===== powers ===== for instance A x B x (A x B x (A x B x C)) ... p = 0: [ (-0.59329, -0.45567) (-0.64778, -0.04562) (+0.24376, -0.60662) (-0.15430, -0.46844) (+0.68713, +0.51889) (+0.24171, +0.12279) (-0.34298, +0.13026) (-0.31959, -0.32133) ] mag: 1.67537 p = 1: [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] mag: 2.74118 p = 2: [ (+2.13861, -0.01369) (+0.54209, -0.89478) (+0.83575, +0.41504) (+1.24944, -1.05630) (-0.04351, -0.46180) (+0.26327, +0.34546) (+0.01041, -1.54119) (+0.40952, -0.34957) ] mag: 3.50500 p = 3: [ (+1.70760, +0.13242) (+0.80526, -0.65975) (-1.02230, +2.73355) (+1.24808, +0.96666) (-1.06437, -1.17383) (+1.76674, +0.61781) (+1.19765, -2.64433) (+0.75502, -0.26786) ] mag: 5.48718 ratio: 5.48718 / 2.74118 = 2.00176 p = 4: [ (-0.67710, +4.22719) (+1.58826, +1.36495) (-1.09467, +1.51355) (+1.67429, +2.81480) (+0.92614, +0.06613) (-0.76879, +0.40610) (+3.03965, +0.52798) (+0.55540, +0.92369) ] mag: 7.01617 ratio: 7.01617 / 3.50500 = 2.00176 p = 5: [ (-0.82368, +3.32806) (+1.03755, +1.80720) (-5.06080, -2.91854) (-2.31960, +2.14606) (+2.66815, -1.71513) (-1.80155, +3.28502) (+4.82692, +3.23538) (+0.28031, +1.57898) ] mag: 10.98403 ratio: 10.98403 / 5.48718 = 2.00176 p = 6: [ (-8.12365, -2.72869) (-3.21801, +2.68662) (-2.62808, -2.65974) (-6.10907, +2.37912) (-0.43550, +1.80690) (-0.54872, -1.65168) (-2.04328, +5.82796) (-2.00668, +0.79253) ] mag: 14.04470 ratio: 14.04470 / 7.01617 = 2.00176 p = 7: [ (-6.30006, -2.72208) (-3.90993, +1.45364) (+7.42888, -9.03165) (-3.47369, -5.28661) (+2.50805, +5.83296) (-5.89312, -4.63872) (-7.97748, +8.46552) (-3.20998, +0.03360) ] mag: 21.98739 ratio: 21.98739 / 10.98403 = 2.00176 for instance A x (A x (A x B x C) x C) x C ... p = 0: [ (+0.02085, +0.40570) (+0.06198, -0.56892) (-0.09484, +0.48290) (+0.38199, -0.43140) (+0.39251, -0.44612) (+0.11080, -0.26022) (-0.12322, +0.68923) (-0.47051, -0.23121) ] mag: 1.50495 p = 1: [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] mag: 2.74118 p = 2: [ (+0.75615, -0.54678) (-0.21472, -0.97180) (+0.15536, +0.28628) (-0.12426, -1.38444) (-0.07872, -1.13580) (+0.30782, -0.52740) (-0.81178, +0.42646) (-0.65601, -0.26802) ] mag: 2.62900 p = 3: [ (-0.61960, -1.27470) (-0.80787, -0.29908) (+2.37945, -0.41325) (+0.16828, -1.29547) (-0.39303, +1.25094) (-0.29774, -1.51993) (-2.38907, +0.25073) (-0.50703, -0.42710) ] mag: 4.54072 ratio: 4.54072 / 2.74118 = 1.65649 p = 4: [ (+0.84724, -1.29284) (-0.90632, -1.37712) (+0.40953, +0.35127) (-1.01007, -2.06916) (-0.79272, -1.71125) (+0.16486, -0.99803) (-1.00440, +1.13949) (-1.17355, -0.02729) ] mag: 4.35490 ratio: 4.35490 / 2.62900 = 1.65649 p = 5: [ (-1.71183, -1.60672) (-1.42692, +0.01435) (+3.43831, -2.04504) (-0.50479, -2.10426) (+0.13066, +2.16809) (-1.35857, -2.17635) (-3.54921, +1.79921) (-1.03696, -0.36148) ] mag: 7.52164 ratio: 7.52164 / 4.54072 = 1.65649 p = 6: [ (+0.54752, -2.50123) (-2.21605, -1.59588) (+0.84128, +0.30172) (-2.78538, -2.60558) (-2.23761, -2.18008) (-0.33439, -1.64192) (-0.88133, +2.35675) (-1.83229, +0.65097) ] mag: 7.21383 ratio: 7.21383 / 4.35490 = 1.65649 p = 7: [ (-3.59828, -1.47540) (-2.19981, +0.86506) (+4.11313, -5.19583) (-2.02416, -2.95836) (+1.48285, +3.27815) (-3.38805, -2.56562) (-4.42996, +4.88090) (-1.81831, +0.05309) ] mag: 12.45948 ratio: 12.45948 / 7.52164 = 1.65649 for instance ((A x B x C) x B x C) x B x C ... p = 0: [ (-0.58095, +0.54826) (-0.43443, -0.14279) (+0.34092, +0.08455) (+0.43339, +0.01640) (+0.69312, +0.65326) (-0.10353, +0.21420) (+0.64615, +0.50118) (-0.28836, -0.11950) ] mag: 1.69958 p = 1: [ (-0.07506, -0.85231) (-0.39126, -0.34259) (+1.43097, +0.27914) (+0.37378, -0.69443) (-0.49096, +0.62092) (+0.15926, -0.92134) (-1.40142, -0.37288) (-0.19403, -0.35003) ] mag: 2.74118 p = 2: [ (-1.36253, -0.46547) (+0.42917, +0.97733) (-1.15220, -0.05549) (-1.30775, +1.75564) (-1.33504, +0.29364) (-0.46272, -0.38386) (-0.42318, +0.80729) (+0.43351, +0.49932) ] mag: 3.58241 p = 3: [ (+1.59782, -0.00244) (+0.70038, -0.67280) (-0.74970, +2.61741) (+1.23182, +0.80721) (-1.07621, -1.01339) (+1.68852, +0.44463) (+0.91934, -2.54734) (+0.68247, -0.30464) ] mag: 5.11906 ratio: 5.11906 / 2.74118 = 1.86747 p = 4: [ (+1.08534, -2.46007) (-1.88743, +0.64115) (+0.28873, -2.13476) (-3.05587, -2.71570) (-0.33142, -2.53113) (+0.78867, -0.79910) (-1.43386, -0.91728) (-0.99877, +0.72617) ] mag: 6.69004 ratio: 6.69004 / 3.58241 = 1.86747 p = 5: [ (-0.25267, +2.97317) (+1.13901, +1.41136) (-4.74905, -1.81618) (-1.70012, +2.16188) (+2.05866, -1.83917) (-1.09905, +3.06995) (+4.59137, +2.12050) (+0.45693, +1.31879) ] mag: 9.55967 ratio: 9.55967 / 5.11906 = 1.86747 p = 6: [ (+4.40229, +2.41531) (-0.88904, -3.61480) (+3.92527, +0.88083) (+5.54453, -5.24833) (+4.76255, -0.20916) (+1.35978, +1.59597) (+1.93744, -2.52006) (-1.19026, -1.97512) ] mag: 12.49342 ratio: 12.49342 / 6.69004 = 1.86747 p = 7: [ (-5.49095, -0.94871) (-2.80922, +1.89195) (+4.14351, -8.54332) (-3.74853, -3.51111) (+3.09041, +4.12624) (-5.53476, -2.53898) (-4.68433, +8.20096) (-2.52719, +0.63783) ] mag: 17.85236 ratio: 17.85236 / 9.55967 = 1.86747