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",[581,1050,1052,1071],{"className":1051},[584],[581,1053,1055],{"className":1054},[588],[590,1056,1057],{"xmlns":592},[594,1058,1059,1068],{},[597,1060,1061],{},[600,1062,1063,1066],{"accent":602},[604,1064,1065],{},"u",[608,1067,610],{},[612,1069,1070],{"encoding":614},"\\vec{u}",[581,1072,1074],{"className":1073,"ariaHidden":602},[619],[581,1075,1077,1080],{"className":1076},[623],[581,1078],{"className":1079,"style":628},[627],[581,1081,1083],{"className":1082},[632,633],[581,1084,1086],{"className":1085},[637],[581,1087,1089],{"className":1088},[641],[581,1090,1092,1100],{"className":1091,"style":628},[645],[581,1093,1094,1097],{"style":648},[581,1095],{"className":1096,"style":653},[652],[581,1098,1065],{"className":1099},[632,657],[581,1101,1102,1105],{"style":648},[581,1103],{"className":1104,"style":653},[652],[581,1106,1108],{"className":1107,"style":871},[666],[581,1109,1111],{"className":1110,"style":672},[671],[674,1112,1113],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,1114],{"d":685},", ",[581,1117,1119,1138],{"className":1118},[584],[581,1120,1122],{"className":1121},[588],[590,1123,1124],{"xmlns":592},[594,1125,1126,1135],{},[597,1127,1128],{},[600,1129,1130,1133],{"accent":602},[604,1131,1132],{},"v",[608,1134,610],{},[612,1136,1137],{"encoding":614},"\\vec{v}",[581,1139,1141],{"className":1140,"ariaHidden":602},[619],[581,1142,1144,1147],{"className":1143},[623],[581,1145],{"className":1146,"style":628},[627],[581,1148,1150],{"className":1149},[632,633],[581,1151,1153],{"className":1152},[637],[581,1154,1156],{"className":1155},[641],[581,1157,1159,1168],{"className":1158,"style":628},[645],[581,1160,1161,1164],{"style":648},[581,1162],{"className":1163,"style":653},[652],[581,1165,1132],{"className":1166,"style":1167},[632,657],"margin-right:0.0359em;",[581,1169,1170,1173],{"style":648},[581,1171],{"className":1172,"style":653},[652],[581,1174,1176],{"className":1175,"style":871},[666],[581,1177,1179],{"className":1178,"style":672},[671],[674,1180,1181],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,1182],{"d":685}," mit dem Kreuzprodukt: ",[581,1185,1187,1222],{"className":1186},[584],[581,1188,1190],{"className":1189},[588],[590,1191,1192],{"xmlns":592},[594,1193,1194,1219],{},[597,1195,1196,1202,1204,1210,1213],{},[600,1197,1198,1200],{"accent":602},[604,1199,606],{},[608,1201,610],{},[608,1203,745],{},[600,1205,1206,1208],{"accent":602},[604,1207,1065],{},[608,1209,610],{},[608,1211,1212],{},"×",[600,1214,1215,1217],{"accent":602},[604,1216,1132],{},[608,1218,610],{},[612,1220,1221],{"encoding":614},"\\vec{n} = \\vec{u} \\times \\vec{v}",[581,1223,1225,1275,1325],{"className":1224,"ariaHidden":602},[619],[581,1226,1228,1231,1266,1269,1272],{"className":1227},[623],[581,1229],{"className":1230,"style":628},[627],[581,1232,1234],{"className":1233},[632,633],[581,1235,1237],{"className":1236},[637],[581,1238,1240],{"className":1239},[641],[581,1241,1243,1251],{"className":1242,"style":628},[645],[581,1244,1245,1248],{"style":648},[581,1246],{"className":1247,"style":653},[652],[581,1249,606],{"className":1250},[632,657],[581,1252,1253,1256],{"style":648},[581,1254],{"className":1255,"style":653},[652],[581,1257,1259],{"className":1258,"style":667},[666],[581,1260,1262],{"className":1261,"style":672},[671],[674,1263,1264],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,1265],{"d":685},[581,1267],{"className":1268,"style":828},[827],[581,1270,745],{"className":1271},[832],[581,1273],{"className":1274,"style":828},[827],[581,1276,1278,1281,1316,1319,1322],{"className":1277},[623],[581,1279],{"className":1280,"style":842},[627],[581,1282,1284],{"className":1283},[632,633],[581,1285,1287],{"className":1286},[637],[581,1288,1290],{"className":1289},[641],[581,1291,1293,1301],{"className":1292,"style":628},[645],[581,1294,1295,1298],{"style":648},[581,1296],{"className":1297,"style":653},[652],[581,1299,1065],{"className":1300},[632,657],[581,1302,1303,1306],{"style":648},[581,1304],{"className":1305,"style":653},[652],[581,1307,1309],{"className":1308,"style":871},[666],[581,1310,1312],{"className":1311,"style":672},[671],[674,1313,1314],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,1315],{"d":685},[581,1317],{"className":1318,"style":882},[827],[581,1320,1212],{"className":1321},[886],[581,1323],{"className":1324,"style":882},[827],[581,1326,1328,1331],{"className":1327},[623],[581,1329],{"className":1330,"style":628},[627],[581,1332,1334],{"className":1333},[632,633],[581,1335,1337],{"className":1336},[637],[581,1338,1340],{"className":1339},[641],[581,1341,1343,1351],{"className":1342,"style":628},[645],[581,1344,1345,1348],{"style":648},[581,1346],{"className":1347,"style":653},[652],[581,1349,1132],{"className":1350,"style":1167},[632,657],[581,1352,1353,1356],{"style":648},[581,1354],{"className":1355,"style":653},[652],[581,1357,1359],{"className":1358,"style":871},[666],[581,1360,1362],{"className":1361,"style":672},[671],[674,1363,1364],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,1365],{"d":685},". Jedes Vielfache ",[581,1368,1370,1394],{"className":1369},[584],[581,1371,1373],{"className":1372},[588],[590,1374,1375],{"xmlns":592},[594,1376,1377,1391],{},[597,1378,1379,1382,1385],{},[604,1380,1381],{},"k",[608,1383,1384],{},"⋅",[600,1386,1387,1389],{"accent":602},[604,1388,606],{},[608,1390,610],{},[612,1392,1393],{"encoding":614},"k \\cdot \\vec{n}",[581,1395,1397,1417],{"className":1396,"ariaHidden":602},[619],[581,1398,1400,1404,1408,1411,1414],{"className":1399},[623],[581,1401],{"className":1402,"style":1403},[627],"height:0.6944em;",[581,1405,1381],{"className":1406,"style":1407},[632,657],"margin-right:0.0315em;",[581,1409],{"className":1410,"style":882},[827],[581,1412,1384],{"className":1413},[886],[581,1415],{"className":1416,"style":882},[827],[581,1418,1420,1423],{"className":1419},[623],[581,1421],{"className":1422,"style":628},[627],[581,1424,1426],{"className":1425},[632,633],[581,1427,1429],{"className":1428},[637],[581,1430,1432],{"className":1431},[641],[581,1433,1435,1443],{"className":1434,"style":628},[645],[581,1436,1437,1440],{"style":648},[581,1438],{"className":1439,"style":653},[652],[581,1441,606],{"className":1442},[632,657],[581,1444,1445,1448],{"style":648},[581,1446],{"className":1447,"style":653},[652],[581,1449,1451],{"className":1450,"style":667},[666],[581,1452,1454],{"className":1453,"style":672},[671],[674,1455,1456],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,1457],{"d":685}," mit ",[581,1460,1462,1482],{"className":1461},[584],[581,1463,1465],{"className":1464},[588],[590,1466,1467],{"xmlns":592},[594,1468,1469,1479],{},[597,1470,1471,1473,1477],{},[604,1472,1381],{},[608,1474,1476],{"mathvariant":1475},"normal","≠",[1027,1478,1029],{},[612,1480,1481],{"encoding":614},"k \\neq 0",[581,1483,1485,1544],{"className":1484,"ariaHidden":602},[619],[581,1486,1488,1492,1495,1498,1541],{"className":1487},[623],[581,1489],{"className":1490,"style":1491},[627],"height:0.8889em;vertical-align:-0.1944em;",[581,1493,1381],{"className":1494,"style":1407},[632,657],[581,1496],{"className":1497,"style":828},[827],[581,1499,1501,1534,1538],{"className":1500},[832],[581,1502,1504],{"className":1503},[832],[581,1505,1508],{"className":1506},[632,1507],"vbox",[581,1509,1512],{"className":1510},[1511],"thinbox",[581,1513,1516,1519,1530],{"className":1514},[1515],"rlap",[581,1517],{"className":1518,"style":1491},[627],[581,1520,1523],{"className":1521},[1522],"inner",[581,1524,1526],{"className":1525},[632],[581,1527,1529],{"className":1528},[832],"",[581,1531],{"className":1532},[1533],"fix",[581,1535],{"className":1536},[827,1537],"nobreak",[581,1539,745],{"className":1540},[832],[581,1542],{"className":1543,"style":828},[827],[581,1545,1547,1550],{"className":1546},[623],[581,1548],{"className":1549,"style":1041},[627],[581,1551,1029],{"className":1552},[632]," ist ebenfalls ein Normalenvektor.",[564,1555,1556,1557,579,1560,2037],{},"Multipliziert man das Skalarprodukt aus, erhält man die ",[568,1558,1559],{},"Koordinatenform",[581,1561,1563,1636],{"className":1562},[584],[581,1564,1566],{"className":1565},[588],[590,1567,1568],{"xmlns":592},[594,1569,1570,1633],{},[597,1571,1572,1580,1586,1589,1596,1602,1604,1611,1617,1619,1625,1627],{},[1573,1574,1575,1577],"msub",{},[604,1576,606],{},[1027,1578,1579],{},"1",[1573,1581,1582,1584],{},[604,1583,750],{},[1027,1585,1579],{},[608,1587,1588],{},"+",[1573,1590,1591,1593],{},[604,1592,606],{},[1027,1594,1595],{},"2",[1573,1597,1598,1600],{},[604,1599,750],{},[1027,1601,1595],{},[608,1603,1588],{},[1573,1605,1606,1608],{},[604,1607,606],{},[1027,1609,1610],{},"3",[1573,1612,1613,1615],{},[604,1614,750],{},[1027,1616,1610],{},[608,1618,745],{},[600,1620,1621,1623],{"accent":602},[604,1622,606],{},[608,1624,610],{},[608,1626,1384],{},[600,1628,1629,1631],{"accent":602},[604,1630,564],{},[608,1632,610],{},[612,1634,1635],{"encoding":614},"n_1 x_1 + n_2 x_2 + n_3 x_3 = \\vec{n} \\cdot \\vec{p}",[581,1637,1639,1744,1839,1935,1985],{"className":1638,"ariaHidden":602},[619],[581,1640,1642,1646,1695,1735,1738,1741],{"className":1641},[623],[581,1643],{"className":1644,"style":1645},[627],"height:0.7333em;vertical-align:-0.15em;",[581,1647,1649,1652],{"className":1648},[632],[581,1650,606],{"className":1651},[632,657],[581,1653,1656],{"className":1654},[1655],"msupsub",[581,1657,1659,1686],{"className":1658},[637,903],[581,1660,1662,1683],{"className":1661},[641],[581,1663,1666],{"className":1664,"style":1665},[645],"height:0.3011em;",[581,1667,1669,1673],{"style":1668},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[581,1670],{"className":1671,"style":1672},[652],"height:2.7em;",[581,1674,1680],{"className":1675},[1676,1677,1678,1679],"sizing","reset-size6","size3","mtight",[581,1681,1579],{"className":1682},[632,1679],[581,1684,938],{"className":1685},[937],[581,1687,1689],{"className":1688},[641],[581,1690,1693],{"className":1691,"style":1692},[645],"height:0.15em;",[581,1694],{},[581,1696,1698,1701],{"className":1697},[632],[581,1699,750],{"className":1700},[632,657],[581,1702,1704],{"className":1703},[1655],[581,1705,1707,1727],{"className":1706},[637,903],[581,1708,1710,1724],{"className":1709},[641],[581,1711,1713],{"className":1712,"style":1665},[645],[581,1714,1715,1718],{"style":1668},[581,1716],{"className":1717,"style":1672},[652],[581,1719,1721],{"className":1720},[1676,1677,1678,1679],[581,1722,1579],{"className":1723},[632,1679],[581,1725,938],{"className":1726},[937],[581,1728,1730],{"className":1729},[641],[581,1731,1733],{"className":1732,"style":1692},[645],[581,1734],{},[581,1736],{"className":1737,"style":882},[827],[581,1739,1588],{"className":1740},[886],[581,1742],{"className":1743,"style":882},[827],[581,1745,1747,1750,1790,1830,1833,1836],{"className":1746},[623],[581,1748],{"className":1749,"style":1645},[627],[581,1751,1753,1756],{"className":1752},[632],[581,1754,606],{"className":1755},[632,657],[581,1757,1759],{"className":1758},[1655],[581,1760,1762,1782],{"className":1761},[637,903],[581,1763,1765,1779],{"className":1764},[641],[581,1766,1768],{"className":1767,"style":1665},[645],[581,1769,1770,1773],{"style":1668},[581,1771],{"className":1772,"style":1672},[652],[581,1774,1776],{"className":1775},[1676,1677,1678,1679],[581,1777,1595],{"className":1778},[632,1679],[581,1780,938],{"className":1781},[937],[581,1783,1785],{"className":1784},[641],[581,1786,1788],{"className":1787,"style":1692},[645],[581,1789],{},[581,1791,1793,1796],{"className":1792},[632],[581,1794,750],{"className":1795},[632,657],[581,1797,1799],{"className":1798},[1655],[581,1800,1802,1822],{"className":1801},[637,903],[581,1803,1805,1819],{"className":1804},[641],[581,1806,1808],{"className":1807,"style":1665},[645],[581,1809,1810,1813],{"style":1668},[581,1811],{"className":1812,"style":1672},[652],[581,1814,1816],{"className":1815},[1676,1677,1678,1679],[581,1817,1595],{"className":1818},[632,1679],[581,1820,938],{"className":1821},[937],[581,1823,1825],{"className":1824},[641],[581,1826,1828],{"className":1827,"style":1692},[645],[581,1829],{},[581,1831],{"className":1832,"style":882},[827],[581,1834,1588],{"className":1835},[886],[581,1837],{"className":1838,"style":882},[827],[581,1840,1842,1846,1886,1926,1929,1932],{"className":1841},[623],[581,1843],{"className":1844,"style":1845},[627],"height:0.5806em;vertical-align:-0.15em;",[581,1847,1849,1852],{"className":1848},[632],[581,1850,606],{"className":1851},[632,657],[581,1853,1855],{"className":1854},[1655],[581,1856,1858,1878],{"className":1857},[637,903],[581,1859,1861,1875],{"className":1860},[641],[581,1862,1864],{"className":1863,"style":1665},[645],[581,1865,1866,1869],{"style":1668},[581,1867],{"className":1868,"style":1672},[652],[581,1870,1872],{"className":1871},[1676,1677,1678,1679],[581,1873,1610],{"className":1874},[632,1679],[581,1876,938],{"className":1877},[937],[581,1879,1881],{"className":1880},[641],[581,1882,1884],{"className":1883,"style":1692},[645],[581,1885],{},[581,1887,1889,1892],{"className":1888},[632],[581,1890,750],{"className":1891},[632,657],[581,1893,1895],{"className":1894},[1655],[581,1896,1898,1918],{"className":1897},[637,903],[581,1899,1901,1915],{"className":1900},[641],[581,1902,1904],{"className":1903,"style":1665},[645],[581,1905,1906,1909],{"style":1668},[581,1907],{"className":1908,"style":1672},[652],[581,1910,1912],{"className":1911},[1676,1677,1678,1679],[581,1913,1610],{"className":1914},[632,1679],[581,1916,938],{"className":1917},[937],[581,1919,1921],{"className":1920},[641],[581,1922,1924],{"className":1923,"style":1692},[645],[581,1925],{},[581,1927],{"className":1928,"style":828},[827],[581,1930,745],{"className":1931},[832],[581,1933],{"className":1934,"style":828},[827],[581,1936,1938,1941,1976,1979,1982],{"className":1937},[623],[581,1939],{"className":1940,"style":628},[627],[581,1942,1944],{"className":1943},[632,633],[581,1945,1947],{"className":1946},[637],[581,1948,1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Ebene durch ",[581,2046,2048,2080],{"className":2047},[584],[581,2049,2051],{"className":2050},[588],[590,2052,2053],{"xmlns":592},[594,2054,2055,2077],{},[597,2056,2057,2059,2063,2065,2068,2070,2072,2074],{},[604,2058,737],{},[608,2060,2062],{"stretchy":2061},"false","(",[1027,2064,1579],{},[608,2066,2067],{},"∣",[1027,2069,1029],{},[608,2071,2067],{},[1027,2073,1595],{},[608,2075,2076],{"stretchy":2061},")",[612,2078,2079],{"encoding":614},"P(1 \\mid 0 \\mid 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",[581,3199,3201,3243],{"className":3200},[584],[581,3202,3204],{"className":3203},[588],[590,3205,3206],{"xmlns":592},[594,3207,3208,3240],{},[597,3209,3210,3216,3218,3224,3226,3228,3230,3232,3234,3236,3238],{},[600,3211,3212,3214],{"accent":602},[604,3213,606],{},[608,3215,610],{},[608,3217,1384],{},[600,3219,3220,3222],{"accent":602},[604,3221,1065],{},[608,3223,610],{},[608,3225,745],{},[1027,3227,2581],{},[608,3229,755],{},[1027,3231,2581],{},[608,3233,1588],{},[1027,3235,1029],{},[608,3237,745],{},[1027,3239,1029],{},[612,3241,3242],{"encoding":614},"\\vec{n} \\cdot \\vec{u} = 6 - 6 + 0 = 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für ",[581,4238,4240,4269],{"className":4239},[584],[581,4241,4243],{"className":4242},[588],[590,4244,4245],{"xmlns":592},[594,4246,4247,4266],{},[597,4248,4249,4252,4254,4256,4258,4260,4262,4264],{},[604,4250,4251],{},"B",[608,4253,2062],{"stretchy":2061},[1027,4255,1610],{},[608,4257,2067],{},[1027,4259,1579],{},[608,4261,2067],{},[1027,4263,1595],{},[608,4265,2076],{"stretchy":2061},[612,4267,4268],{"encoding":614},"B(3 \\mid 1 \\mid 2)",[581,4270,4272,4297,4315],{"className":4271,"ariaHidden":602},[619],[581,4273,4275,4278,4282,4285,4288,4291,4294],{"className":4274},[623],[581,4276],{"className":4277,"style":2089},[627],[581,4279,4251],{"className":4280,"style":4281},[632,657],"margin-right:0.0502em;",[581,4283,2062],{"className":4284},[2096],[581,4286,1610],{"className":4287},[632],[581,4289],{"className":4290,"style":828},[827],[581,4292,2067],{"className":4293},[832],[581,4295],{"className":4296,"style":828},[827],[581,4298,4300,4303,4306,4309,4312],{"className":4299},[623],[581,4301],{"className":4302,"style":2089},[627],[581,4304,1579],{"className":4305},[632],[581,4307],{"className":4308,"style":828},[827],[581,4310,2067],{"className":4311},[832],[581,4313],{"className":4314,"style":828},[827],[581,4316,4318,4321,4324],{"className":4317},[623],[581,4319],{"className":4320,"style":2089},[627],[581,4322,1595],{"className":4323},[632],[581,4325,2076],{"className":4326},[2139],": ",[581,4329,4331,4399],{"className":4330},[584],[581,4332,4334],{"className":4333},[588],[590,4335,4336],{"xmlns":592},[594,4337,4338,4396],{},[597,4339,4340,4342,4344,4346,4348,4350,4352,4354,4356,4358,4360,4362,4364,4366,4368,4370,4372,4374,4376,4378,4380,4382,4384,4386,4388,4390,4392,4394],{},[608,4341,2062],{"stretchy":2061},[1027,4343,1595],{},[608,4345,2167],{"separator":602},[2169,4347,2171],{},[1027,4349,1579],{},[608,4351,2167],{"separator":602},[2169,4353,2171],{},[1027,4355,1029],{},[608,4357,2076],{"stretchy":2061},[608,4359,1384],{},[608,4361,2062],{"stretchy":2061},[1027,4363,1610],{},[608,4365,2167],{"separator":602},[2169,4367,2171],{},[608,4369,755],{},[1027,4371,2581],{},[608,4373,2167],{"separator":602},[2169,4375,2171],{},[1027,4377,2590],{},[608,4379,2076],{"stretchy":2061},[608,4381,745],{},[1027,4383,2581],{},[608,4385,755],{},[1027,4387,2581],{},[608,4389,1588],{},[1027,4391,1029],{},[608,4393,745],{},[1027,4395,1029],{},[612,4397,4398],{"encoding":614},"(2,\\ 1,\\ 0) \\cdot (3,\\ -6,\\ 5) = 6 - 6 + 0 = 0",[581,4400,4402,4450,4501,4519,4537,4555],{"className":4401,"ariaHidden":602},[619],[581,4403,4405,4408,4411,4414,4417,4420,4423,4426,4429,4432,4435,4438,4441,4444,4447],{"className":4404},[623],[581,4406],{"className":4407,"style":2089},[627],[581,4409,2062],{"className":4410},[2096],[581,4412,1595],{"className":4413},[632],[581,4415,2167],{"className":4416},[2253],[581,4418,2171],{"className":4419},[827],[581,4421],{"className":4422,"style":2260},[827],[581,4424,1579],{"className":4425},[632],[581,4427,2167],{"className":4428},[2253],[581,4430,2171],{"className":4431},[827],[581,4433],{"className":4434,"style":2260},[827],[581,4436,1029],{"className":4437},[632],[581,4439,2076],{"className":4440},[2139],[581,4442],{"className":4443,"style":882},[827],[581,4445,1384],{"className":4446},[886],[581,4448],{"className":4449,"style":882},[827],[581,4451,4453,4456,4459,4462,4465,4468,4471,4474,4477,4480,4483,4486,4489,4492,4495,4498],{"className":4452},[623],[581,4454],{"className":4455,"style":2089},[627],[581,4457,2062],{"className":4458},[2096],[581,4460,1610],{"className":4461},[632],[581,4463,2167],{"className":4464},[2253],[581,4466,2171],{"className":4467},[827],[581,4469],{"className":4470,"style":2260},[827],[581,4472,755],{"className":4473},[632],[581,4475,2581],{"className":4476},[632],[581,4478,2167],{"className":4479},[2253],[581,4481,2171],{"className":4482},[827],[581,4484],{"className":4485,"style":2260},[827],[581,4487,2590],{"className":4488},[632],[581,4490,2076],{"className":4491},[2139],[581,4493],{"className":4494,"style":828},[827],[581,4496,745],{"className":4497},[832],[581,4499],{"className":4500,"style":828},[827],[581,4502,4504,4507,4510,4513,4516],{"className":4503},[623],[581,4505],{"className":4506,"style":3352},[627],[581,4508,2581],{"className":4509},[632],[581,4511],{"className":4512,"style":882},[827],[581,4514,755],{"className":4515},[886],[581,4517],{"className":4518,"style":882},[827],[581,4520,4522,4525,4528,4531,4534],{"className":4521},[623],[581,4523],{"className":4524,"style":3352},[627],[581,4526,2581],{"className":4527},[632],[581,4529],{"className":4530,"style":882},[827],[581,4532,1588],{"className":4533},[886],[581,4535],{"className":4536,"style":882},[827],[581,4538,4540,4543,4546,4549,4552],{"className":4539},[623],[581,4541],{"className":4542,"style":1041},[627],[581,4544,1029],{"className":4545},[632],[581,4547],{"className":4548,"style":828},[827],[581,4550,745],{"className":4551},[832],[581,4553],{"className":4554,"style":828},[827],[581,4556,4558,4561],{"className":4557},[623],[581,4559],{"className":4560,"style":1041},[627],[581,4562,1029],{"className":4563},[632],[581,4565,4567,4580],{"className":4566},[584],[581,4568,4570],{"className":4569},[588],[590,4571,4572],{"xmlns":592},[594,4573,4574,4578],{},[597,4575,4576],{},[604,4577,4251],{},[612,4579,4251],{"encoding":614},[581,4581,4583],{"className":4582,"ariaHidden":602},[619],[581,4584,4586,4589],{"className":4585},[623],[581,4587],{"className":4588,"style":713},[627],[581,4590,4251],{"className":4591,"style":4281},[632,657]," liegt in ",[581,4594,4596,4609],{"className":4595},[584],[581,4597,4599],{"className":4598},[588],[590,4600,4601],{"xmlns":592},[594,4602,4603,4607],{},[597,4604,4605],{},[604,4606,3646],{},[612,4608,3646],{"encoding":614},[581,4610,4612],{"className":4611,"ariaHidden":602},[619],[581,4613,4615,4618],{"className":4614},[623],[581,4616],{"className":4617,"style":713},[627],[581,4619,3646],{"className":4620,"style":3768},[632,657],". Für ",[581,4623,4625,4654],{"className":4624},[584],[581,4626,4628],{"className":4627},[588],[590,4629,4630],{"xmlns":592},[594,4631,4632,4651],{},[597,4633,4634,4637,4639,4641,4643,4645,4647,4649],{},[604,4635,4636],{},"Q",[608,4638,2062],{"stretchy":2061},[1027,4640,1595],{},[608,4642,2067],{},[1027,4644,1029],{},[608,4646,2067],{},[1027,4648,1610],{},[608,4650,2076],{"stretchy":2061},[612,4652,4653],{"encoding":614},"Q(2 \\mid 0 \\mid 3)",[581,4655,4657,4681,4699],{"className":4656,"ariaHidden":602},[619],[581,4658,4660,4663,4666,4669,4672,4675,4678],{"className":4659},[623],[581,4661],{"className":4662,"style":2089},[627],[581,4664,4636],{"className":4665},[632,657],[581,4667,2062],{"className":4668},[2096],[581,4670,1595],{"className":4671},[632],[581,4673],{"className":4674,"style":828},[827],[581,4676,2067],{"className":4677},[832],[581,4679],{"className":4680,"style":828},[827],[581,4682,4684,4687,4690,4693,4696],{"className":4683},[623],[581,4685],{"className":4686,"style":2089},[627],[581,4688,1029],{"className":4689},[632],[581,4691],{"className":4692,"style":828},[827],[581,4694,2067],{"className":4695},[832],[581,4697],{"className":4698,"style":828},[827],[581,4700,4702,4705,4708],{"className":4701},[623],[581,4703],{"className":4704,"style":2089},[627],[581,4706,1610],{"className":4707},[632],[581,4709,2076],{"className":4710},[2139],[581,4712,4714,4783],{"className":4713},[584],[581,4715,4717],{"className":4716},[588],[590,4718,4719],{"xmlns":592},[594,4720,4721,4780],{},[597,4722,4723,4725,4727,4729,4731,4733,4735,4737,4739,4741,4743,4745,4747,4749,4751,4753,4755,4757,4759,4761,4763,4765,4767,4769,4771,4773,4776,4778],{},[608,4724,2062],{"stretchy":2061},[1027,4726,1579],{},[608,4728,2167],{"separator":602},[2169,4730,2171],{},[1027,4732,1029],{},[608,4734,2167],{"separator":602},[2169,4736,2171],{},[1027,4738,1579],{},[608,4740,2076],{"stretchy":2061},[608,4742,1384],{},[608,4744,2062],{"stretchy":2061},[1027,4746,1610],{},[608,4748,2167],{"separator":602},[2169,4750,2171],{},[608,4752,755],{},[1027,4754,2581],{},[608,4756,2167],{"separator":602},[2169,4758,2171],{},[1027,4760,2590],{},[608,4762,2076],{"stretchy":2061},[608,4764,745],{},[1027,4766,1610],{},[608,4768,1588],{},[1027,4770,2590],{},[608,4772,745],{},[1027,4774,4775],{},"8",[608,4777,1476],{"mathvariant":1475},[1027,4779,1029],{},[612,4781,4782],{"encoding":614},"(1,\\ 0,\\ 1) \\cdot (3,\\ -6,\\ 5) = 3 + 5 = 8 \\neq 0",[581,4784,4786,4834,4885,4903,4921,4972],{"className":4785,"ariaHidden":602},[619],[581,4787,4789,4792,4795,4798,4801,4804,4807,4810,4813,4816,4819,4822,4825,4828,4831],{"className":4788},[623],[581,4790],{"className":4791,"style":2089},[627],[581,4793,2062],{"className":4794},[2096],[581,4796,1579],{"className":4797},[632],[581,4799,2167],{"className":4800},[2253],[581,4802,2171],{"className":4803},[827],[581,4805],{"className":4806,"style":2260},[827],[581,4808,1029],{"className":4809},[632],[581,4811,2167],{"className":4812},[2253],[581,4814,2171],{"className":4815},[827],[581,4817],{"className":4818,"style":2260},[827],[581,4820,1579],{"className":4821},[632],[581,4823,2076],{"className":4824},[2139],[581,4826],{"className":4827,"style":882},[827],[581,4829,1384],{"className":4830},[886],[581,4832],{"className":4833,"style":882},[827],[581,4835,4837,4840,4843,4846,4849,4852,4855,4858,4861,4864,4867,4870,4873,4876,4879,4882],{"className":4836},[623],[581,4838],{"className":4839,"style":2089},[627],[581,4841,2062],{"className":4842},[2096],[581,4844,1610],{"className":4845},[632],[581,4847,2167],{"className":4848},[2253],[581,4850,2171],{"className":4851},[827],[581,4853],{"className":4854,"style":2260},[827],[581,4856,755],{"className":4857},[632],[581,4859,2581],{"className":4860},[632],[581,4862,2167],{"className":4863},[2253],[581,4865,2171],{"className":4866},[827],[581,4868],{"className":4869,"style":2260},[827],[581,4871,2590],{"className":4872},[632],[581,4874,2076],{"className":4875},[2139],[581,4877],{"className":4878,"style":828},[827],[581,4880,745],{"className":4881},[832],[581,4883],{"className":4884,"style":828},[827],[581,4886,4888,4891,4894,4897,4900],{"className":4887},[623],[581,4889],{"className":4890,"style":3352},[627],[581,4892,1610],{"className":4893},[632],[581,4895],{"className":4896,"style":882},[827],[581,4898,1588],{"className":4899},[886],[581,4901],{"className":4902,"style":882},[827],[581,4904,4906,4909,4912,4915,4918],{"className":4905},[623],[581,4907],{"className":4908,"style":1041},[627],[581,4910,2590],{"className":4911},[632],[581,4913],{"className":4914,"style":828},[827],[581,4916,745],{"className":4917},[832],[581,4919],{"className":4920,"style":828},[827],[581,4922,4924,4927,4930,4933,4969],{"className":4923},[623],[581,4925],{"className":4926,"style":1491},[627],[581,4928,4775],{"className":4929},[632],[581,4931],{"className":4932,"style":828},[827],[581,4934,4936,4963,4966],{"className":4935},[832],[581,4937,4939],{"className":4938},[832],[581,4940,4942],{"className":4941},[632,1507],[581,4943,4945],{"className":4944},[1511],[581,4946,4948,4951,4960],{"className":4947},[1515],[581,4949],{"className":4950,"style":1491},[627],[581,4952,4954],{"className":4953},[1522],[581,4955,4957],{"className":4956},[632],[581,4958,1529],{"className":4959},[832],[581,4961],{"className":4962},[1533],[581,4964],{"className":4965},[827,1537],[581,4967,745],{"className":4968},[832],[581,4970],{"className":4971,"style":828},[827],[581,4973,4975,4978],{"className":4974},[623],[581,4976],{"className":4977,"style":1041},[627],[581,4979,1029],{"className":4980},[632],[581,4982,4984,4997],{"className":4983},[584],[581,4985,4987],{"className":4986},[588],[590,4988,4989],{"xmlns":592},[594,4990,4991,4995],{},[597,4992,4993],{},[604,4994,4636],{},[612,4996,4636],{"encoding":614},[581,4998,5000],{"className":4999,"ariaHidden":602},[619],[581,5001,5003,5007],{"className":5002},[623],[581,5004],{"className":5005,"style":5006},[627],"height:0.8778em;vertical-align:-0.1944em;",[581,5008,4636],{"className":5009},[632,657]," liegt nicht in ",[581,5012,5014,5027],{"className":5013},[584],[581,5015,5017],{"className":5016},[588],[590,5018,5019],{"xmlns":592},[594,5020,5021,5025],{},[597,5022,5023],{},[604,5024,3646],{},[612,5026,3646],{"encoding":614},[581,5028,5030],{"className":5029,"ariaHidden":602},[619],[581,5031,5033,5036],{"className":5032},[623],[581,5034],{"className":5035,"style":713},[627],[581,5037,3646],{"className":5038,"style":3768},[632,657],[559,5040,5042],{"id":5041},"typische-fehler","Typische Fehler",[5044,5045,5046,5053,5230],"ul",{},[5047,5048,5049,5050,5052],"li",{},"Einen Spannvektor als Normalenvektor verwenden. Der Normalenvektor steht ",[568,5051,574],{}," auf der Ebene, nicht in ihr.",[5047,5054,5055,5056,5201,5202,2037],{},"Die Klammer weglassen und ",[581,5057,5059,5089],{"className":5058},[584],[581,5060,5062],{"className":5061},[588],[590,5063,5064],{"xmlns":592},[594,5065,5066,5086],{},[597,5067,5068,5074,5076,5082,5084],{},[600,5069,5070,5072],{"accent":602},[604,5071,750],{},[608,5073,610],{},[608,5075,1384],{},[600,5077,5078,5080],{"accent":602},[604,5079,606],{},[608,5081,610],{},[608,5083,745],{},[1027,5085,1029],{},[612,5087,5088],{"encoding":614},"\\vec{x} \\cdot \\vec{n} = 0",[581,5090,5092,5142,5192],{"className":5091,"ariaHidden":602},[619],[581,5093,5095,5098,5133,5136,5139],{"className":5094},[623],[581,5096],{"className":5097,"style":628},[627],[581,5099,5101],{"className":5100},[632,633],[581,5102,5104],{"className":5103},[637],[581,5105,5107],{"className":5106},[641],[581,5108,5110,5118],{"className":5109,"style":628},[645],[581,5111,5112,5115],{"style":648},[581,5113],{"className":5114,"style":653},[652],[581,5116,750],{"className":5117},[632,657],[581,5119,5120,5123],{"style":648},[581,5121],{"className":5122,"style":653},[652],[581,5124,5126],{"className":5125,"style":871},[666],[581,5127,5129],{"className":5128,"style":672},[671],[674,5130,5131],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,5132],{"d":685},[581,5134],{"className":5135,"style":882},[827],[581,5137,1384],{"className":5138},[886],[581,5140],{"className":5141,"style":882},[827],[581,5143,5145,5148,5183,5186,5189],{"className":5144},[623],[581,5146],{"className":5147,"style":628},[627],[581,5149,5151],{"className":5150},[632,633],[581,5152,5154],{"className":5153},[637],[581,5155,5157],{"className":5156},[641],[581,5158,5160,5168],{"className":5159,"style":628},[645],[581,5161,5162,5165],{"style":648},[581,5163],{"className":5164,"style":653},[652],[581,5166,606],{"className":5167},[632,657],[581,5169,5170,5173],{"style":648},[581,5171],{"className":5172,"style":653},[652],[581,5174,5176],{"className":5175,"style":667},[666],[581,5177,5179],{"className":5178,"style":672},[671],[674,5180,5181],{"xmlns":676,"width":677,"height":678,"style":679,"viewBox":680,"preserveAspectRatio":681},[683,5182],{"d":685},[581,5184],{"className":5185,"style":828},[827],[581,5187,745],{"className":5188},[832],[581,5190],{"className":5191,"style":828},[827],[581,5193,5195,5198],{"className":5194},[623],[581,5196],{"className":5197,"style":1041},[627],[581,5199,1029],{"className":5200},[632]," schreiben. Das beschreibt eine Ebene durch den Ursprung, nicht durch ",[581,5203,5205,5218],{"className":5204},[584],[581,5206,5208],{"className":5207},[588],[590,5209,5210],{"xmlns":592},[594,5211,5212,5216],{},[597,5213,5214],{},[604,5215,737],{},[612,5217,737],{"encoding":614},[581,5219,5221],{"className":5220,"ariaHidden":602},[619],[581,5222,5224,5227],{"className":5223},[623],[581,5225],{"className":5226,"style":713},[627],[581,5228,737],{"className":5229,"style":798},[632,657],[5047,5231,5232],{},"Beim Kreuzprodukt das Vorzeichen der mittleren Koordinate vergessen und so einen Vektor erhalten, der nicht senkrecht steht. Die Probe mit dem Skalarprodukt deckt das auf.",{"title":5234,"searchDepth":5235,"depth":5235,"links":5236},"",2,[5237,5238,5239],{"id":561,"depth":5235,"text":562},{"id":2040,"depth":5235,"text":2041},{"id":5041,"depth":5235,"text":5042},[5241,5244,5246,5249,5251,5253,5255,5257,5259],{"lhs":5242,"rhs":5243},"(x1 - p1)*n1 + (x2 - p2)*n2 + (x3 - p3)*n3","n1*x1 + n2*x2 + n3*x3 - (n1*p1 + n2*p2 + n3*p3)",{"lhs":5245,"rhs":1610},"1*3 - 0*2",{"lhs":5247,"rhs":5248},"0*(-1) - 2*3","-6",{"lhs":5250,"rhs":2590},"2*2 - 1*(-1)",{"lhs":5252,"rhs":1029},"3*2 + (-6)*1 + 5*0",{"lhs":5254,"rhs":1029},"3*(-1) + (-6)*2 + 5*3",{"lhs":5256,"rhs":1029},"(3 - 1)*3 + (1 - 0)*(-6) + (2 - 2)*5",{"lhs":5258,"rhs":1029},"(0 - 1)*3 + (2 - 0)*(-6) + (5 - 2)*5",{"lhs":5260,"rhs":4775},"(2 - 1)*3 + (0 - 0)*(-6) + (3 - 2)*5","\\vec{n} \\neq \\vec{0}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath 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style=\"height:0.1944em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mclose\">)\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003Cspan class=\"mbin\">⋅\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2222em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.714em;\">\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.714em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">n\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.2355em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.471em\" height=\"0.714em\" style=\"width:0.471em\" viewBox=\"0 0 471 714\" preserveAspectRatio=\"xMinYMin\">\u003Cpath d=\"M377 20c0-5.333 1.833-10 5.5-14S391 0 397 0c4.667 0 8.667 1.667 12 5\n3.333 2.667 6.667 9 10 19 6.667 24.667 20.333 43.667 41 57 7.333 4.667 11\n10.667 11 18 0 6-1 10-3 12s-6.667 5-14 9c-28.667 14.667-53.667 35.667-75 63\n-1.333 1.333-3.167 3.5-5.5 6.5s-4 4.833-5 5.5c-1 .667-2.5 1.333-4.5 2s-4.333 1\n-7 1c-4.667 0-9.167-1.833-13.5-5.5S337 184 337 178c0-12.667 15.667-32.333 47-59\nH213l-171-1c-8.667-6-13-12.333-13-19 0-4.667 4.333-11.333 13-20h359\nc-16-25.333-24-45-24-59z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6444em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fnormalenform-ebene",[],[5274,5275,5276,5277],"plane-coordinate-form","plane-parametric-form","cross-product","orthogonal-vectors",{"title":548,"description":5234},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fnormalenform-ebene","Ein Punkt liegt in der Ebene, wenn der Vektor vom Stützpunkt zu ihm senkrecht auf dem Normalenvektor steht.",[5282,5283,5284,5285,5286],"ebene","normalenform","normalenvektor","skalarprodukt","analytische geometrie",[5288,5291,5295,5299],{"symbol":3646,"meaning":5289,"symbolHtml":5290},"Name der Ebene","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>E\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">E\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:0.6833em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":5292,"meaning":5293,"symbolHtml":5294},"\\vec{x}","Ortsvektor eines beliebigen Punkts X der Ebene","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover 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