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als Zeilen.",[905,1165,1166,1167,1196,1197,1200],{},"Ist es ",[580,1168,1170,1184],{"className":1169},[583],[580,1171,1173],{"className":1172},[587],[589,1174,1175],{"xmlns":591},[593,1176,1177,1182],{},[596,1178,1179],{},[925,1180,1181],{},"0",[627,1183,1181],{"encoding":629},[580,1185,1187],{"className":1186,"ariaHidden":606},[634],[580,1188,1190,1193],{"className":1189},[638],[580,1191],{"className":1192,"style":963},[642],[580,1194,1181],{"className":1195},[647],", liegen die drei Vektoren in einer Ebene (sie sind ",[568,1198,1199],{},"linear abhängig","). Damit prüft man, ob vier Punkte in einer Ebene liegen.",[905,1202,1203,1204,1207,1208,575],{},"Eine dreiseitige Pyramide (",[568,1205,1206],{},"Tetraeder","), die von denselben Vektoren aufgespannt wird, hat das Volumen ",[580,1209,1211,1274],{"className":1210},[583],[580,1212,1214],{"className":1213},[587],[589,1215,1216],{"xmlns":591},[593,1217,1218,1271],{},[596,1219,1220,1223,1226,1235],{},[599,1221,1222],{},"V",[611,1224,1225],{},"=",[1227,1228,1229,1232],"mfrac",{},[925,1230,1231],{},"1",[925,1233,1234],{},"6",[596,1236,1237,1239,1243,1249,1251,1257,1260,1263,1269],{},[611,1238,602],{"fence":606},[611,1240,1242],{"stretchy":1241},"false","(",[604,1244,1245,1247],{"accent":606},[599,1246,609],{},[611,1248,613],{},[611,1250,616],{},[604,1252,1253,1255],{"accent":606},[599,1254,621],{},[611,1256,613],{},[611,1258,1259],{"stretchy":1241},")",[611,1261,1262],{},"⋅",[604,1264,1265,1267],{"accent":606},[599,1266,781],{},[611,1268,613],{},[611,1270,602],{"fence":606},[627,1272,1273],{"encoding":629},"V = \\frac{1}{6} \\left|(\\vec{a} \\times \\vec{b}) \\cdot \\vec{c}\\right|",[580,1275,1277,1298],{"className":1276,"ariaHidden":606},[634],[580,1278,1280,1284,1287,1291,1295],{"className":1279},[638],[580,1281],{"className":1282,"style":1283},[642],"height:0.6833em;",[580,1285,1222],{"className":1286,"style":709},[647,676],[580,1288],{"className":1289,"style":1290},[708],"margin-right:0.2778em;",[580,1292,1225],{"className":1293},[1294],"mrel",[580,1296],{"className":1297,"style":1290},[708],[580,1299,1301,1305,1390,1394],{"className":1300},[638],[580,1302],{"className":1303,"style":1304},[642],"height:1.8em;vertical-align:-0.65em;",[580,1306,1308,1313,1386],{"className":1307},[647],[580,1309],{"className":1310},[1311,1312],"mopen","nulldelimiter",[580,1314,1316],{"className":1315},[1227],[580,1317,1320,1377],{"className":1318},[655,1319],"vlist-t2",[580,1321,1323,1372],{"className":1322},[659],[580,1324,1327,1346,1357],{"className":1325,"style":1326},[663],"height:0.8451em;",[580,1328,1330,1333],{"style":1329},"top:-2.655em;",[580,1331],{"className":1332,"style":672},[671],[580,1334,1340],{"className":1335},[1336,1337,1338,1339],"sizing","reset-size6","size3","mtight",[580,1341,1343],{"className":1342},[647,1339],[580,1344,1234],{"className":1345},[647,1339],[580,1347,1349,1352],{"style":1348},"top:-3.23em;",[580,1350],{"className":1351,"style":672},[671],[580,1353],{"className":1354,"style":1356},[1355],"frac-line","border-bottom-width:0.04em;",[580,1358,1360,1363],{"style":1359},"top:-3.394em;",[580,1361],{"className":1362,"style":672},[671],[580,1364,1366],{"className":1365},[1336,1337,1338,1339],[580,1367,1369],{"className":1368},[647,1339],[580,1370,1231],{"className":1371},[647,1339],[580,1373,1376],{"className":1374},[1375],"vlist-s","​",[580,1378,1380],{"className":1379},[659],[580,1381,1384],{"className":1382,"style":1383},[663],"height:0.345em;",[580,1385],{},[580,1387],{"className":1388},[1389,1312],"mclose",[580,1391],{"className":1392,"style":1393},[708],"margin-right:0.1667em;",[580,1395,1398,1446,1449,1484,1487,1490,1493,1528,1531,1534,1537,1540,1575],{"className":1396},[1397],"minner",[580,1399,1401],{"className":1400},[1311],[580,1402,1406],{"className":1403},[1404,1405],"delimsizing","mult",[580,1407,1409,1437],{"className":1408},[655,1319],[580,1410,1412,1434],{"className":1411},[659],[580,1413,1416],{"className":1414,"style":1415},[663],"height:1.15em;",[580,1417,1419,1423],{"style":1418},"top:-3.15em;",[580,1420],{"className":1421,"style":1422},[671],"height:3.8em;",[580,1424,1426],{"style":1425},"width:0.333em;height:1.8em;",[693,1427,1431],{"xmlns":695,"width":1428,"height":1429,"viewBox":1430},"0.333em","1.8em","0 0 333 1800",[702,1432],{"d":1433},"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z",[580,1435,1376],{"className":1436},[1375],[580,1438,1440],{"className":1439},[659],[580,1441,1444],{"className":1442,"style":1443},[663],"height:0.65em;",[580,1445],{},[580,1447,1242],{"className":1448},[1311],[580,1450,1452],{"className":1451},[647,651],[580,1453,1455],{"className":1454},[655],[580,1456,1458],{"className":1457},[659],[580,1459,1461,1469],{"className":1460,"style":664},[663],[580,1462,1463,1466],{"style":667},[580,1464],{"className":1465,"style":672},[671],[580,1467,609],{"className":1468},[647,676],[580,1470,1471,1474],{"style":667},[580,1472],{"className":1473,"style":672},[671],[580,1475,1477],{"className":1476,"style":686},[685],[580,1478,1480],{"className":1479,"style":691},[690],[693,1481,1482],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,1483],{"d":704},[580,1485],{"className":1486,"style":709},[708],[580,1488,616],{"className":1489},[713],[580,1491],{"className":1492,"style":709},[708],[580,1494,1496],{"className":1495},[647,651],[580,1497,1499],{"className":1498},[655],[580,1500,1502],{"className":1501},[659],[580,1503,1505,1513],{"className":1504,"style":736},[663],[580,1506,1507,1510],{"style":667},[580,1508],{"className":1509,"style":672},[671],[580,1511,621],{"className":1512},[647,676],[580,1514,1515,1518],{"style":747},[580,1516],{"className":1517,"style":672},[671],[580,1519,1521],{"className":1520,"style":686},[685],[580,1522,1524],{"className":1523,"style":691},[690],[693,1525,1526],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,1527],{"d":704},[580,1529,1259],{"className":1530},[1389],[580,1532],{"className":1533,"style":709},[708],[580,1535,1262],{"className":1536},[713],[580,1538],{"className":1539,"style":709},[708],[580,1541,1543],{"className":1542},[647,651],[580,1544,1546],{"className":1545},[655],[580,1547,1549],{"className":1548},[659],[580,1550,1552,1560],{"className":1551,"style":664},[663],[580,1553,1554,1557],{"style":667},[580,1555],{"className":1556,"style":672},[671],[580,1558,781],{"className":1559},[647,676],[580,1561,1562,1565],{"style":667},[580,1563],{"className":1564,"style":672},[671],[580,1566,1568],{"className":1567,"style":824},[685],[580,1569,1571],{"className":1570,"style":691},[690],[693,1572,1573],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,1574],{"d":704},[580,1576,1578],{"className":1577},[1389],[580,1579,1581],{"className":1580},[1404,1405],[580,1582,1584,1604],{"className":1583},[655,1319],[580,1585,1587,1601],{"className":1586},[659],[580,1588,1590],{"className":1589,"style":1415},[663],[580,1591,1592,1595],{"style":1418},[580,1593],{"className":1594,"style":1422},[671],[580,1596,1597],{"style":1425},[693,1598,1599],{"xmlns":695,"width":1428,"height":1429,"viewBox":1430},[702,1600],{"d":1433},[580,1602,1376],{"className":1603},[1375],[580,1605,1607],{"className":1606},[659],[580,1608,1610],{"className":1609,"style":1443},[663],[580,1611],{},[559,1613,1615],{"id":1614},"beispiele","Beispiele",[564,1617,1618,1619,1033,1757,1890,1891,2025,2026,2218],{},"Für 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0)",[580,1665,1667,1717],{"className":1666,"ariaHidden":606},[634],[580,1668,1670,1673,1708,1711,1714],{"className":1669},[638],[580,1671],{"className":1672,"style":664},[642],[580,1674,1676],{"className":1675},[647,651],[580,1677,1679],{"className":1678},[655],[580,1680,1682],{"className":1681},[659],[580,1683,1685,1693],{"className":1684,"style":664},[663],[580,1686,1687,1690],{"style":667},[580,1688],{"className":1689,"style":672},[671],[580,1691,609],{"className":1692},[647,676],[580,1694,1695,1698],{"style":667},[580,1696],{"className":1697,"style":672},[671],[580,1699,1701],{"className":1700,"style":686},[685],[580,1702,1704],{"className":1703,"style":691},[690],[693,1705,1706],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,1707],{"d":704},[580,1709],{"className":1710,"style":1290},[708],[580,1712,1225],{"className":1713},[1294],[580,1715],{"className":1716,"style":1290},[708],[580,1718,1720,1723,1726,1729,1733,1736,1739,1742,1745,1748,1751,1754],{"className":1719},[638],[580,1721],{"className":1722,"style":643},[642],[580,1724,1242],{"className":1725},[1311],[580,1727,1643],{"className":1728},[647],[580,1730,1646],{"className":1731},[1732],"mpunct",[580,1734,1650],{"className":1735},[708],[580,1737],{"className":1738,"style":1393},[708],[580,1740,1181],{"className":1741},[647],[580,1743,1646],{"className":1744},[1732],[580,1746,1650],{"className":1747},[708],[580,1749],{"className":1750,"style":1393},[708],[580,1752,1181],{"className":1753},[647],[580,1755,1259],{"className":1756},[1389],[580,1758,1760,1798],{"className":1759},[583],[580,1761,1763],{"className":1762},[587],[589,1764,1765],{"xmlns":591},[593,1766,1767,1795],{},[596,1768,1769,1775,1777,1779,1781,1783,1785,1787,1789,1791,1793],{},[604,1770,1771,1773],{"accent":606},[599,1772,621],{},[611,1774,613],{},[611,1776,1225],{},[611,1778,1242],{"stretchy":1241},[925,1780,1231],{},[611,1782,1646],{"separator":606},[1648,1784,1650],{},[925,1786,927],{},[611,1788,1646],{"separator":606},[1648,1790,1650],{},[925,1792,1181],{},[611,1794,1259],{"stretchy":1241},[627,1796,1797],{"encoding":629},"\\vec{b} = (1,\\ 3,\\ 0)",[580,1799,1801,1851],{"className":1800,"ariaHidden":606},[634],[580,1802,1804,1807,1842,1845,1848],{"className":1803},[638],[580,1805],{"className":1806,"style":736},[642],[580,1808,1810],{"className":1809},[647,651],[580,1811,1813],{"className":1812},[655],[580,1814,1816],{"className":1815},[659],[580,1817,1819,1827],{"className":1818,"style":736},[663],[580,1820,1821,1824],{"style":667},[580,1822],{"className":1823,"style":672},[671],[580,1825,621],{"className":1826},[647,676],[580,1828,1829,1832],{"style":747},[580,1830],{"className":1831,"style":672},[671],[580,1833,1835],{"className":1834,"style":686},[685],[580,1836,1838],{"className":1837,"style":691},[690],[693,1839,1840],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,1841],{"d":704},[580,1843],{"className":1844,"style":1290},[708],[580,1846,1225],{"className":1847},[1294],[580,1849],{"className":1850,"style":1290},[708],[580,1852,1854,1857,1860,1863,1866,1869,1872,1875,1878,1881,1884,1887],{"className":1853},[638],[580,1855],{"className":1856,"style":643},[642],[580,1858,1242],{"className":1859},[1311],[580,1861,1231],{"className":1862},[647],[580,1864,1646],{"className":1865},[1732],[580,1867,1650],{"className":1868},[708],[580,1870],{"className":1871,"style":1393},[708],[580,1873,927],{"className":1874},[647],[580,1876,1646],{"className":1877},[1732],[580,1879,1650],{"className":1880},[708],[580,1882],{"className":1883,"style":1393},[708],[580,1885,1181],{"className":1886},[647],[580,1888,1259],{"className":1889},[1389]," und ",[580,1892,1894,1933],{"className":1893},[583],[580,1895,1897],{"className":1896},[587],[589,1898,1899],{"xmlns":591},[593,1900,1901,1930],{},[596,1902,1903,1909,1911,1913,1915,1917,1919,1921,1923,1925,1928],{},[604,1904,1905,1907],{"accent":606},[599,1906,781],{},[611,1908,613],{},[611,1910,1225],{},[611,1912,1242],{"stretchy":1241},[925,1914,1181],{},[611,1916,1646],{"separator":606},[1648,1918,1650],{},[925,1920,1231],{},[611,1922,1646],{"separator":606},[1648,1924,1650],{},[925,1926,1927],{},"4",[611,1929,1259],{"stretchy":1241},[627,1931,1932],{"encoding":629},"\\vec{c} = (0,\\ 1,\\ 4)",[580,1934,1936,1986],{"className":1935,"ariaHidden":606},[634],[580,1937,1939,1942,1977,1980,1983],{"className":1938},[638],[580,1940],{"className":1941,"style":664},[642],[580,1943,1945],{"className":1944},[647,651],[580,1946,1948],{"className":1947},[655],[580,1949,1951],{"className":1950},[659],[580,1952,1954,1962],{"className":1953,"style":664},[663],[580,1955,1956,1959],{"style":667},[580,1957],{"className":1958,"style":672},[671],[580,1960,781],{"className":1961},[647,676],[580,1963,1964,1967],{"style":667},[580,1965],{"className":1966,"style":672},[671],[580,1968,1970],{"className":1969,"style":824},[685],[580,1971,1973],{"className":1972,"style":691},[690],[693,1974,1975],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,1976],{"d":704},[580,1978],{"className":1979,"style":1290},[708],[580,1981,1225],{"className":1982},[1294],[580,1984],{"className":1985,"style":1290},[708],[580,1987,1989,1992,1995,1998,2001,2004,2007,2010,2013,2016,2019,2022],{"className":1988},[638],[580,1990],{"className":1991,"style":643},[642],[580,1993,1242],{"className":1994},[1311],[580,1996,1181],{"className":1997},[647],[580,1999,1646],{"className":2000},[1732],[580,2002,1650],{"className":2003},[708],[580,2005],{"className":2006,"style":1393},[708],[580,2008,1231],{"className":2009},[647],[580,2011,1646],{"className":2012},[1732],[580,2014,1650],{"className":2015},[708],[580,2017],{"className":2018,"style":1393},[708],[580,2020,1927],{"className":2021},[647],[580,2023,1259],{"className":2024},[1389]," ist ",[580,2027,2029,2075],{"className":2028},[583],[580,2030,2032],{"className":2031},[587],[589,2033,2034],{"xmlns":591},[593,2035,2036,2072],{},[596,2037,2038,2044,2046,2052,2054,2056,2058,2060,2062,2064,2066,2068,2070],{},[604,2039,2040,2042],{"accent":606},[599,2041,609],{},[611,2043,613],{},[611,2045,616],{},[604,2047,2048,2050],{"accent":606},[599,2049,621],{},[611,2051,613],{},[611,2053,1225],{},[611,2055,1242],{"stretchy":1241},[925,2057,1181],{},[611,2059,1646],{"separator":606},[1648,2061,1650],{},[925,2063,1181],{},[611,2065,1646],{"separator":606},[1648,2067,1650],{},[925,2069,1234],{},[611,2071,1259],{"stretchy":1241},[627,2073,2074],{"encoding":629},"\\vec{a} \\times \\vec{b} = (0,\\ 0,\\ 6)",[580,2076,2078,2129,2179],{"className":2077,"ariaHidden":606},[634],[580,2079,2081,2085,2120,2123,2126],{"className":2080},[638],[580,2082],{"className":2083,"style":2084},[642],"height:0.7973em;vertical-align:-0.0833em;",[580,2086,2088],{"className":2087},[647,651],[580,2089,2091],{"className":2090},[655],[580,2092,2094],{"className":2093},[659],[580,2095,2097,2105],{"className":2096,"style":664},[663],[580,2098,2099,2102],{"style":667},[580,2100],{"className":2101,"style":672},[671],[580,2103,609],{"className":2104},[647,676],[580,2106,2107,2110],{"style":667},[580,2108],{"className":2109,"style":672},[671],[580,2111,2113],{"className":2112,"style":686},[685],[580,2114,2116],{"className":2115,"style":691},[690],[693,2117,2118],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,2119],{"d":704},[580,2121],{"className":2122,"style":709},[708],[580,2124,616],{"className":2125},[713],[580,2127],{"className":2128,"style":709},[708],[580,2130,2132,2135,2170,2173,2176],{"className":2131},[638],[580,2133],{"className":2134,"style":736},[642],[580,2136,2138],{"className":2137},[647,651],[580,2139,2141],{"className":2140},[655],[580,2142,2144],{"className":2143},[659],[580,2145,2147,2155],{"className":2146,"style":736},[663],[580,2148,2149,2152],{"style":667},[580,2150],{"className":2151,"style":672},[671],[580,2153,621],{"className":2154},[647,676],[580,2156,2157,2160],{"style":747},[580,2158],{"className":2159,"style":672},[671],[580,2161,2163],{"className":2162,"style":686},[685],[580,2164,2166],{"className":2165,"style":691},[690],[693,2167,2168],{"xmlns":695,"width":696,"height":697,"style":698,"viewBox":699,"preserveAspectRatio":700},[702,2169],{"d":704},[580,2171],{"className":2172,"style":1290},[708],[580,2174,1225],{"className":2175},[1294],[580,2177],{"className":2178,"style":1290},[708],[580,2180,2182,2185,2188,2191,2194,2197,2200,2203,2206,2209,2212,2215],{"className":2181},[638],[580,2183],{"className":2184,"style":643},[642],[580,2186,1242],{"className":2187},[1311],[580,2189,1181],{"className":2190},[647],[580,2192,1646],{"className":2193},[1732],[580,2195,1650],{"className":2196},[708],[580,2198],{"className":2199,"style":1393},[708],[580,2201,1181],{"className":2202},[647],[580,2204,1646],{"className":2205},[1732],[580,2207,1650],{"className":2208},[708],[580,2210],{"className":2211,"style":1393},[708],[580,2213,1234],{"className":2214},[647],[580,2216,1259],{"className":2217},[1389],", also",[580,2220,2223],{"className":2221},[2222],"katex-display",[580,2224,2226,2275],{"className":2225},[583],[580,2227,2229],{"className":2228},[587],[589,2230,2232],{"xmlns":591,"display":2231},"block",[593,2233,2234,2272],{},[596,2235,2236,2238,2240,2242,2244,2246,2248,2251,2253,2255,2257,2259,2261,2263,2265,2267,2269],{},[599,2237,1222],{},[611,2239,1225],{},[599,2241,602],{"mathvariant":601},[925,2243,1181],{},[611,2245,1262],{},[925,2247,1181],{},[611,2249,2250],{},"+",[925,2252,1181],{},[611,2254,1262],{},[925,2256,1231],{},[611,2258,2250],{},[925,2260,1234],{},[611,2262,1262],{},[925,2264,1927],{},[599,2266,602],{"mathvariant":601},[611,2268,1225],{},[925,2270,2271],{},"24",[627,2273,2274],{"encoding":629},"V = |0 \\cdot 0 + 0 \\cdot 1 + 6 \\cdot 4| = 24",[580,2276,2278,2296,2315,2333,2351,2369,2387,2406],{"className":2277,"ariaHidden":606},[634],[580,2279,2281,2284,2287,2290,2293],{"className":2280},[638],[580,2282],{"className":2283,"style":1283},[642],[580,2285,1222],{"className":2286,"style":709},[647,676],[580,2288],{"className":2289,"style":1290},[708],[580,2291,1225],{"className":2292},[1294],[580,2294],{"className":2295,"style":1290},[708],[580,2297,2299,2302,2306,2309,2312],{"className":2298},[638],[580,2300],{"className":2301,"style":643},[642],[580,2303,2305],{"className":2304},[647],"∣0",[580,2307],{"className":2308,"style":709},[708],[580,2310,1262],{"className":2311},[713],[580,2313],{"className":2314,"style":709},[708],[580,2316,2318,2321,2324,2327,2330],{"className":2317},[638],[580,2319],{"className":2320,"style":944},[642],[580,2322,1181],{"className":2323},[647],[580,2325],{"className":2326,"style":709},[708],[580,2328,2250],{"className":2329},[713],[580,2331],{"className":2332,"style":709},[708],[580,2334,2336,2339,2342,2345,2348],{"className":2335},[638],[580,2337],{"className":2338,"style":963},[642],[580,2340,1181],{"className":2341},[647],[580,2343],{"className":2344,"style":709},[708],[580,2346,1262],{"className":2347},[713],[580,2349],{"className":2350,"style":709},[708],[580,2352,2354,2357,2360,2363,2366],{"className":2353},[638],[580,2355],{"className":2356,"style":944},[642],[580,2358,1231],{"className":2359},[647],[580,2361],{"className":2362,"style":709},[708],[580,2364,2250],{"className":2365},[713],[580,2367],{"className":2368,"style":709},[708],[580,2370,2372,2375,2378,2381,2384],{"className":2371},[638],[580,2373],{"className":2374,"style":963},[642],[580,2376,1234],{"className":2377},[647],[580,2379],{"className":2380,"style":709},[708],[580,2382,1262],{"className":2383},[713],[580,2385],{"className":2386,"style":709},[708],[580,2388,2390,2393,2397,2400,2403],{"className":2389},[638],[580,2391],{"className":2392,"style":643},[642],[580,2394,2396],{"className":2395},[647],"4∣",[580,2398],{"className":2399,"style":1290},[708],[580,2401,1225],{"className":2402},[1294],[580,2404],{"className":2405,"style":1290},[708],[580,2407,2409,2412],{"className":2408},[638],[580,2410],{"className":2411,"style":963},[642],[580,2413,2271],{"className":2414},[647],[564,2416,2417,2418,2539,2540,2682,2683,2711,2712,575],{},"Die dreiseitige Pyramide mit denselben Kantenvektoren hat das Volumen ",[580,2419,2421,2443],{"className":2420},[583],[580,2422,2424],{"className":2423},[587],[589,2425,2426],{"xmlns":591},[593,2427,2428,2440],{},[596,2429,2430,2436,2438],{},[1227,2431,2432,2434],{},[925,2433,2271],{},[925,2435,1234],{},[611,2437,1225],{},[925,2439,1927],{},[627,2441,2442],{"encoding":629},"\\frac{24}{6} = 4",[580,2444,2446,2530],{"className":2445,"ariaHidden":606},[634],[580,2447,2449,2453,2521,2524,2527],{"className":2448},[638],[580,2450],{"className":2451,"style":2452},[642],"height:1.1901em;vertical-align:-0.345em;",[580,2454,2456,2459,2518],{"className":2455},[647],[580,2457],{"className":2458},[1311,1312],[580,2460,2462],{"className":2461},[1227],[580,2463,2465,2510],{"className":2464},[655,1319],[580,2466,2468,2507],{"className":2467},[659],[580,2469,2471,2485,2493],{"className":2470,"style":1326},[663],[580,2472,2473,2476],{"style":1329},[580,2474],{"className":2475,"style":672},[671],[580,2477,2479],{"className":2478},[1336,1337,1338,1339],[580,2480,2482],{"className":2481},[647,1339],[580,2483,1234],{"className":2484},[647,1339],[580,2486,2487,2490],{"style":1348},[580,2488],{"className":2489,"style":672},[671],[580,2491],{"className":2492,"style":1356},[1355],[580,2494,2495,2498],{"style":1359},[580,2496],{"className":2497,"style":672},[671],[580,2499,2501],{"className":2500},[1336,1337,1338,1339],[580,2502,2504],{"className":2503},[647,1339],[580,2505,2271],{"className":2506},[647,1339],[580,2508,1376],{"className":2509},[1375],[580,2511,2513],{"className":2512},[659],[580,2514,2516],{"className":2515,"style":1383},[663],[580,2517],{},[580,2519],{"className":2520},[1389,1312],[580,2522],{"className":2523,"style":1290},[708],[580,2525,1225],{"className":2526},[1294],[580,2528],{"className":2529,"style":1290},[708],[580,2531,2533,2536],{"className":2532},[638],[580,2534],{"className":2535,"style":963},[642],[580,2537,1927],{"className":2538},[647],". Probe mit „ein Drittel Grundfläche mal Höhe“: Das Grunddreieck hat die Fläche ",[580,2541,2543,2569],{"className":2542},[583],[580,2544,2546],{"className":2545},[587],[589,2547,2548],{"xmlns":591},[593,2549,2550,2566],{},[596,2551,2552,2558,2560,2562,2564],{},[1227,2553,2554,2556],{},[925,2555,1231],{},[925,2557,1643],{},[611,2559,1262],{},[925,2561,1234],{},[611,2563,1225],{},[925,2565,927],{},[627,2567,2568],{"encoding":629},"\\frac{1}{2} \\cdot 6 = 3",[580,2570,2572,2655,2673],{"className":2571,"ariaHidden":606},[634],[580,2573,2575,2578,2646,2649,2652],{"className":2574},[638],[580,2576],{"className":2577,"style":2452},[642],[580,2579,2581,2584,2643],{"className":2580},[647],[580,2582],{"className":2583},[1311,1312],[580,2585,2587],{"className":2586},[1227],[580,2588,2590,2635],{"className":2589},[655,1319],[580,2591,2593,2632],{"className":2592},[659],[580,2594,2596,2610,2618],{"className":2595,"style":1326},[663],[580,2597,2598,2601],{"style":1329},[580,2599],{"className":2600,"style":672},[671],[580,2602,2604],{"className":2603},[1336,1337,1338,1339],[580,2605,2607],{"className":2606},[647,1339],[580,2608,1643],{"className":2609},[647,1339],[580,2611,2612,2615],{"style":1348},[580,2613],{"className":2614,"style":672},[671],[580,2616],{"className":2617,"style":1356},[1355],[580,2619,2620,2623],{"style":1359},[580,2621],{"className":2622,"style":672},[671],[580,2624,2626],{"className":2625},[1336,1337,1338,1339],[580,2627,2629],{"className":2628},[647,1339],[580,2630,1231],{"className":2631},[647,1339],[580,2633,1376],{"className":2634},[1375],[580,2636,2638],{"className":2637},[659],[580,2639,2641],{"className":2640,"style":1383},[663],[580,2642],{},[580,2644],{"className":2645},[1389,1312],[580,2647],{"className":2648,"style":709},[708],[580,2650,1262],{"className":2651},[713],[580,2653],{"className":2654,"style":709},[708],[580,2656,2658,2661,2664,2667,2670],{"className":2657},[638],[580,2659],{"className":2660,"style":963},[642],[580,2662,1234],{"className":2663},[647],[580,2665],{"className":2666,"style":1290},[708],[580,2668,1225],{"className":2669},[1294],[580,2671],{"className":2672,"style":1290},[708],[580,2674,2676,2679],{"className":2675},[638],[580,2677],{"className":2678,"style":963},[642],[580,2680,927],{"className":2681},[647],", die Höhe ist ",[580,2684,2686,2699],{"className":2685},[583],[580,2687,2689],{"className":2688},[587],[589,2690,2691],{"xmlns":591},[593,2692,2693,2697],{},[596,2694,2695],{},[925,2696,1927],{},[627,2698,1927],{"encoding":629},[580,2700,2702],{"className":2701,"ariaHidden":606},[634],[580,2703,2705,2708],{"className":2704},[638],[580,2706],{"className":2707,"style":963},[642],[580,2709,1927],{"className":2710},[647],", also ",[580,2713,2715,2745],{"className":2714},[583],[580,2716,2718],{"className":2717},[587],[589,2719,2720],{"xmlns":591},[593,2721,2722,2742],{},[596,2723,2724,2730,2732,2734,2736,2738,2740],{},[1227,2725,2726,2728],{},[925,2727,1231],{},[925,2729,927],{},[611,2731,1262],{},[925,2733,927],{},[611,2735,1262],{},[925,2737,1927],{},[611,2739,1225],{},[925,2741,1927],{},[627,2743,2744],{"encoding":629},"\\frac{1}{3} \\cdot 3 \\cdot 4 = 4",[580,2746,2748,2831,2849,2867],{"className":2747,"ariaHidden":606},[634],[580,2749,2751,2754,2822,2825,2828],{"className":2750},[638],[580,2752],{"className":2753,"style":2452},[642],[580,2755,2757,2760,2819],{"className":2756},[647],[580,2758],{"className":2759},[1311,1312],[580,2761,2763],{"className":2762},[1227],[580,2764,2766,2811],{"className":2765},[655,1319],[580,2767,2769,2808],{"className":2768},[659],[580,2770,2772,2786,2794],{"className":2771,"style":1326},[663],[580,2773,2774,2777],{"style":1329},[580,2775],{"className":2776,"style":672},[671],[580,2778,2780],{"className":2779},[1336,1337,1338,1339],[580,2781,2783],{"className":2782},[647,1339],[580,2784,927],{"className":2785},[647,1339],[580,2787,2788,2791],{"style":1348},[580,2789],{"className":2790,"style":672},[671],[580,2792],{"className":2793,"style":1356},[1355],[580,2795,2796,2799],{"style":1359},[580,2797],{"className":2798,"style":672},[671],[580,2800,2802],{"className":2801},[1336,1337,1338,1339],[580,2803,2805],{"className":2804},[647,1339],[580,2806,1231],{"className":2807},[647,1339],[580,2809,1376],{"className":2810},[1375],[580,2812,2814],{"className":2813},[659],[580,2815,2817],{"className":2816,"style":1383},[663],[580,2818],{},[580,2820],{"className":2821},[1389,1312],[580,2823],{"className":2824,"style":709},[708],[580,2826,1262],{"className":2827},[713],[580,2829],{"className":2830,"style":709},[708],[580,2832,2834,2837,2840,2843,2846],{"className":2833},[638],[580,2835],{"className":2836,"style":963},[642],[580,2838,927],{"className":2839},[647],[580,2841],{"className":2842,"style":709},[708],[580,2844,1262],{"className":2845},[713],[580,2847],{"className":2848,"style":709},[708],[580,2850,2852,2855,2858,2861,2864],{"className":2851},[638],[580,2853],{"className":2854,"style":963},[642],[580,2856,1927],{"className":2857},[647],[580,2859],{"className":2860,"style":1290},[708],[580,2862,1225],{"className":2863},[1294],[580,2865],{"className":2866,"style":1290},[708],[580,2868,2870,2873],{"className":2869},[638],[580,2871],{"className":2872,"style":963},[642],[580,2874,1927],{"className":2875},[647],[559,2877,2879],{"id":2878},"typische-fehler","Typische Fehler",[902,2881,2882,2885,3243],{},[905,2883,2884],{},"Die Betragsstriche weglassen. Das Spatprodukt kann negativ sein, ein Volumen nicht.",[905,2886,2887,2888,2916,2917,2945,2946,3044,3045,3143,3144,3242],{},"Beim Tetraeder durch ",[580,2889,2891,2904],{"className":2890},[583],[580,2892,2894],{"className":2893},[587],[589,2895,2896],{"xmlns":591},[593,2897,2898,2902],{},[596,2899,2900],{},[925,2901,927],{},[627,2903,927],{"encoding":629},[580,2905,2907],{"className":2906,"ariaHidden":606},[634],[580,2908,2910,2913],{"className":2909},[638],[580,2911],{"className":2912,"style":963},[642],[580,2914,927],{"className":2915},[647]," statt durch ",[580,2918,2920,2933],{"className":2919},[583],[580,2921,2923],{"className":2922},[587],[589,2924,2925],{"xmlns":591},[593,2926,2927,2931],{},[596,2928,2929],{},[925,2930,1234],{},[627,2932,1234],{"encoding":629},[580,2934,2936],{"className":2935,"ariaHidden":606},[634],[580,2937,2939,2942],{"className":2938},[638],[580,2940],{"className":2941,"style":963},[642],[580,2943,1234],{"className":2944},[647]," teilen. Der Faktor ",[580,2947,2949,2967],{"className":2948},[583],[580,2950,2952],{"className":2951},[587],[589,2953,2954],{"xmlns":591},[593,2955,2956,2964],{},[596,2957,2958],{},[1227,2959,2960,2962],{},[925,2961,1231],{},[925,2963,1234],{},[627,2965,2966],{"encoding":629},"\\frac{1}{6}",[580,2968,2970],{"className":2969,"ariaHidden":606},[634],[580,2971,2973,2976],{"className":2972},[638],[580,2974],{"className":2975,"style":2452},[642],[580,2977,2979,2982,3041],{"className":2978},[647],[580,2980],{"className":2981},[1311,1312],[580,2983,2985],{"className":2984},[1227],[580,2986,2988,3033],{"className":2987},[655,1319],[580,2989,2991,3030],{"className":2990},[659],[580,2992,2994,3008,3016],{"className":2993,"style":1326},[663],[580,2995,2996,2999],{"style":1329},[580,2997],{"className":2998,"style":672},[671],[580,3000,3002],{"className":3001},[1336,1337,1338,1339],[580,3003,3005],{"className":3004},[647,1339],[580,3006,1234],{"className":3007},[647,1339],[580,3009,3010,3013],{"style":1348},[580,3011],{"className":3012,"style":672},[671],[580,3014],{"className":3015,"style":1356},[1355],[580,3017,3018,3021],{"style":1359},[580,3019],{"className":3020,"style":672},[671],[580,3022,3024],{"className":3023},[1336,1337,1338,1339],[580,3025,3027],{"className":3026},[647,1339],[580,3028,1231],{"className":3029},[647,1339],[580,3031,1376],{"className":3032},[1375],[580,3034,3036],{"className":3035},[659],[580,3037,3039],{"className":3038,"style":1383},[663],[580,3040],{},[580,3042],{"className":3043},[1389,1312]," setzt sich aus 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ist nicht definiert, weil ",[580,3439,3441,3467],{"className":3440},[583],[580,3442,3444],{"className":3443},[587],[589,3445,3446],{"xmlns":591},[593,3447,3448,3464],{},[596,3449,3450,3456,3458],{},[604,3451,3452,3454],{"accent":606},[599,3453,621],{},[611,3455,613],{},[611,3457,1262],{},[604,3459,3460,3462],{"accent":606},[599,3461,781],{},[611,3463,613],{},[627,3465,3466],{"encoding":629},"\\vec{b} \\cdot 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eine Zahl ist.",{"title":3563,"searchDepth":3564,"depth":3564,"links":3565},"",2,[3566,3567,3568],{"id":561,"depth":3564,"text":562},{"id":1614,"depth":3564,"text":1615},{"id":2878,"depth":3564,"text":2879},[3570,3573,3575,3577,3579,3581,3583],{"lhs":3571,"rhs":3572},"(a2*b3 - a3*b2)*c1 + (a3*b1 - a1*b3)*c2 + (a1*b2 - a2*b1)*c3","det([[a1, a2, a3], [b1, b2, b3], [c1, c2, c3]])",{"lhs":3571,"rhs":3574},"(b2*c3 - b3*c2)*a1 + (b3*c1 - b1*c3)*a2 + (b1*c2 - b2*c1)*a3",{"lhs":3576,"rhs":1234},"2*3 - 0*1",{"lhs":3578,"rhs":2271},"0*0 + 0*1 + 6*4",{"lhs":3580,"rhs":2271},"det([[2, 0, 0], [1, 3, 0], [0, 1, 4]])",{"lhs":3582,"rhs":1927},"24 \u002F 6",{"lhs":3584,"rhs":1927},"1\u002F3 * (1\u002F2 * 6) * 4","\\vec{a}, \\vec{b}, \\vec{c} \\in \\mathbb{R}^3","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover 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style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">b\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2634em;\">\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=\"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>\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\">c\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"accent-body\" style=\"left:-0.1799em;\">\u003Cspan class=\"katex-overlay\" style=\"height:0.714em;width:0.471em;\">\u003Csvg 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style=\"height:1.15em;\">\u003Cspan style=\"top:-3.15em;\">\u003Cspan class=\"pstrut\" style=\"height:3.8em;\">\u003C\u002Fspan>\u003Cspan style=\"width:0.333em;height:1.8em;\">\u003Csvg xmlns=\"http:\u002F\u002Fwww.w3.org\u002F2000\u002Fsvg\" width=\"0.333em\" height=\"1.8em\" viewBox=\"0 0 333 1800\">\u003Cpath d=\"M145 15 v585 v600 v585 c2.667,10,9.667,15,21,15\nc10,0,16.667,-5,20,-15 v-585 v-600 v-585 c-2.667,-10,-9.667,-15,-21,-15\nc-10,0,-16.667,5,-20,15z M188 15 H145 v585 v600 v585 h43z\"\u002F>\u003C\u002Fsvg>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-s\">​\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.65em;\">\u003Cspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>","upper",{},true,"\u002Fde\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspatprodukt",[],[3598,3599,3600,3601],"cross-product","dot-product","determinant-3x3","parallelogram-area-vectors",{"title":548,"description":3563},"de\u002Fformulas\u002Fmath\u002Flinear-algebra\u002Fspatprodukt","Das Volumen des von drei Vektoren aufgespannten Spats ist der Betrag ihres Spatprodukts.",[3606,3607,3608,3609,3610],"spat","volumen","kreuzprodukt","determinante","vektor",[3612,3615,3619],{"symbol":1222,"meaning":3613,"symbolHtml":3614},"Volumen des Spats","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>V\u003C\u002Fmi>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">V\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:0.2222em;\">V\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",{"symbol":3616,"meaning":3617,"symbolHtml":3618},"\\vec{a}, \\vec{b}, \\vec{c}","drei Vektoren, die von einer gemeinsamen Ecke ausgehen und den Spat aufspannen","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>a\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>b\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>c\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\vec{a}, \\vec{b}, \\vec{c}\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:1.1719em;vertical-align:-0.1944em;\">\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 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