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|\\vec{a}|^2",[579,1077,1079,1129,1179],{"className":1078,"ariaHidden":605},[673],[579,1080,1082,1085,1120,1123,1126],{"className":1081},[677],[579,1083],{"className":1084,"style":682},[681],[579,1086,1088],{"className":1087},[686,687],[579,1089,1091],{"className":1090},[691],[579,1092,1094],{"className":1093},[695],[579,1095,1097,1105],{"className":1096,"style":682},[699],[579,1098,1099,1102],{"style":702},[579,1100],{"className":1101,"style":707},[706],[579,1103,609],{"className":1104},[686,711],[579,1106,1107,1110],{"style":702},[579,1108],{"className":1109,"style":707},[706],[579,1111,1113],{"className":1112,"style":721},[720],[579,1114,1116],{"className":1115,"style":726},[725],[728,1117,1118],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,1119],{"d":739},[579,1121],{"className":1122,"style":744},[743],[579,1124,616],{"className":1125},[748],[579,1127],{"className":1128,"style":744},[743],[579,1130,1132,1135,1170,1173,1176],{"className":1131},[677],[579,1133],{"className":1134,"style":682},[681],[579,1136,1138],{"className":1137},[686,687],[579,1139,1141],{"className":1140},[691],[579,1142,1144],{"className":1143},[695],[579,1145,1147,1155],{"className":1146,"style":682},[699],[579,1148,1149,1152],{"style":702},[579,1150],{"className":1151,"style":707},[706],[579,1153,609],{"className":1154},[686,711],[579,1156,1157,1160],{"style":702},[579,1158],{"className":1159,"style":707},[706],[579,1161,1163],{"className":1162,"style":721},[720],[579,1164,1166],{"className":1165,"style":726},[725],[728,1167,1168],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,1169],{"d":739},[579,1171],{"className":1172,"style":798},[743],[579,1174,626],{"className":1175},[802],[579,1177],{"className":1178,"style":798},[743],[579,1180,1182,1186,1189,1224],{"className":1181},[677],[579,1183],{"className":1184,"style":1185},[681],"height:1.0641em;vertical-align:-0.25em;",[579,1187,630],{"className":1188},[686],[579,1190,1192],{"className":1191},[686,687],[579,1193,1195],{"className":1194},[691],[579,1196,1198],{"className":1197},[695],[579,1199,1201,1209],{"className":1200,"style":682},[699],[579,1202,1203,1206],{"style":702},[579,1204],{"className":1205,"style":707},[706],[579,1207,609],{"className":1208},[686,711],[579,1210,1211,1214],{"style":702},[579,1212],{"className":1213,"style":707},[706],[579,1215,1217],{"className":1216,"style":721},[720],[579,1218,1220],{"className":1219,"style":726},[725],[728,1221,1222],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,1223],{"d":739},[579,1225,1227,1230],{"className":1226},[686],[579,1228,630],{"className":1229},[686],[579,1231,1234],{"className":1232},[1233],"msupsub",[579,1235,1237],{"className":1236},[691],[579,1238,1240],{"className":1239},[695],[579,1241,1244],{"className":1242,"style":1243},[699],"height:0.8141em;",[579,1245,1247,1251],{"style":1246},"top:-3.063em;margin-right:0.05em;",[579,1248],{"className":1249,"style":1250},[706],"height:2.7em;",[579,1252,1258],{"className":1253},[1254,1255,1256,1257],"sizing","reset-size6","size3","mtight",[579,1259,1072],{"className":1260},[686,1257],".",[564,1263,1264],{},"Das Vorzeichen verrät die Lage: positiv bei spitzem Winkel, negativ bei stumpfem Winkel.",[559,1266,1268],{"id":1267},"beispiele","Beispiele",[579,1270,1272],{"className":1271},[582],[579,1273,1275,1423],{"className":1274},[586],[579,1276,1278],{"className":1277},[590],[592,1279,1280],{"xmlns":594,"display":595},[597,1281,1282,1420],{},[600,1283,1284,1328,1330,1370,1372,1374,1376,1378,1381,1383,1385,1387,1389,1391,1393,1395,1397,1399,1401,1403,1405,1407,1410,1412,1415,1417],{},[600,1285,1286,1289,1325],{},[611,1287,1288],{"fence":605},"(",[1290,1291,1295,1308,1316],"mtable",{"rowspacing":1292,"columnalign":1293,"columnspacing":1294},"0.16em","center","1em",[1296,1297,1298],"mtr",{},[1299,1300,1301],"mtd",{},[1302,1303,1305],"mstyle",{"scriptlevel":1006,"displaystyle":1304},"false",[1004,1306,1307],{},"1",[1296,1309,1310],{},[1299,1311,1312],{},[1302,1313,1314],{"scriptlevel":1006,"displaystyle":1304},[1004,1315,1072],{},[1296,1317,1318],{},[1299,1319,1320],{},[1302,1321,1322],{"scriptlevel":1006,"displaystyle":1304},[1004,1323,1324],{},"3",[611,1326,1327],{"fence":605},")",[611,1329,616],{},[600,1331,1332,1334,1368],{},[611,1333,1288],{"fence":605},[1290,1335,1336,1345,1359],{"rowspacing":1292,"columnalign":1293,"columnspacing":1294},[1296,1337,1338],{},[1299,1339,1340],{},[1302,1341,1342],{"scriptlevel":1006,"displaystyle":1304},[1004,1343,1344],{},"4",[1296,1346,1347],{},[1299,1348,1349],{},[1302,1350,1351],{"scriptlevel":1006,"displaystyle":1304},[600,1352,1353,1356],{},[611,1354,1355],{},"−",[1004,1357,1358],{},"5",[1296,1360,1361],{},[1299,1362,1363],{},[1302,1364,1365],{"scriptlevel":1006,"displaystyle":1304},[1004,1366,1367],{},"6",[611,1369,1327],{"fence":605},[611,1371,626],{},[1004,1373,1307],{},[611,1375,616],{},[1004,1377,1344],{},[611,1379,1380],{},"+",[1004,1382,1072],{},[611,1384,616],{},[611,1386,1288],{"stretchy":1304},[611,1388,1355],{},[1004,1390,1358],{},[611,1392,1327],{"stretchy":1304},[611,1394,1380],{},[1004,1396,1324],{},[611,1398,616],{},[1004,1400,1367],{},[611,1402,626],{},[1004,1404,1344],{},[611,1406,1355],{},[1004,1408,1409],{},"10",[611,1411,1380],{},[1004,1413,1414],{},"18",[611,1416,626],{},[1004,1418,1419],{},"12",[666,1421,1422],{"encoding":668},"\\begin{pmatrix} 1 \\\\ 2 \\\\ 3 \\end{pmatrix} \\cdot \\begin{pmatrix} 4 \\\\ -5 \\\\ 6 \\end{pmatrix} = 1 \\cdot 4 + 2 \\cdot (-5) + 3 \\cdot 6 = 4 - 10 + 18 = 12",[579,1424,1426,1603,1760,1778,1797,1815,1842,1860,1878,1896,1914,1932],{"className":1425,"ariaHidden":605},[673],[579,1427,1429,1433,1594,1597,1600],{"className":1428},[677],[579,1430],{"className":1431,"style":1432},[681],"height:3.6em;vertical-align:-1.55em;",[579,1434,1437,1489,1555],{"className":1435},[1436],"minner",[579,1438,1441],{"className":1439},[1440],"mopen",[579,1442,1446],{"className":1443},[1444,1445],"delimsizing","mult",[579,1447,1450,1480],{"className":1448},[691,1449],"vlist-t2",[579,1451,1453,1475],{"className":1452},[695],[579,1454,1457],{"className":1455,"style":1456},[699],"height:2.05em;",[579,1458,1460,1464],{"style":1459},"top:-4.05em;",[579,1461],{"className":1462,"style":1463},[706],"height:5.6em;",[579,1465,1467],{"style":1466},"width:0.875em;height:3.6em;",[728,1468,1472],{"xmlns":730,"width":1469,"height":1470,"viewBox":1471},"0.875em","3.6em","0 0 875 3600",[737,1473],{"d":1474},"M863,9c0,-2,-2,-5,-6,-9c0,0,-17,0,-17,0c-12.7,0,-19.3,0.3,-20,1\nc-5.3,5.3,-10.3,11,-15,17c-242.7,294.7,-395.3,682,-458,1162c-21.3,163.3,-33.3,349,\n-36,557 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:1733,"style":1463},[706],[579,1735,1736],{"style":1466},[728,1737,1738],{"xmlns":730,"width":1469,"height":1470,"viewBox":1471},[737,1739],{"d":1582},[579,1741,1479],{"className":1742},[1478],[579,1744,1746],{"className":1745},[695],[579,1747,1749],{"className":1748,"style":1486},[699],[579,1750],{},[579,1752],{"className":1753,"style":798},[743],[579,1755,626],{"className":1756},[802],[579,1758],{"className":1759,"style":798},[743],[579,1761,1763,1766,1769,1772,1775],{"className":1762},[677],[579,1764],{"className":1765,"style":1018},[681],[579,1767,1307],{"className":1768},[686],[579,1770],{"className":1771,"style":744},[743],[579,1773,616],{"className":1774},[748],[579,1776],{"className":1777,"style":744},[743],[579,1779,1781,1785,1788,1791,1794],{"className":1780},[677],[579,1782],{"className":1783,"style":1784},[681],"height:0.7278em;vertical-align:-0.0833em;",[579,1786,1344],{"className":1787},[686],[579,1789],{"className":1790,"style":744},[743],[579,1792,1380],{"className":1793},[748],[579,1795],{"className":1796,"style":744},[743],[579,1798,1800,1803,1806,1809,1812],{"className":1799},[677],[579,1801],{"className":1802,"style":1018},[681],[579,1804,1072],{"className":1805},[686],[579,1807],{"className":1808,"style":744},[743],[579,1810,616],{"className":1811},[748],[579,1813],{"className":1814,"style":744},[743],[579,1816,1818,1821,1824,1827,1830,1833,1836,1839],{"className":1817},[677],[579,1819],{"className":1820,"style":812},[681],[579,1822,1288],{"className":1823},[1440],[579,1825,1355],{"className":1826},[686],[579,1828,1358],{"className":1829},[686],[579,1831,1327],{"className":1832},[1558],[579,1834],{"className":1835,"style":744},[743],[579,1837,1380],{"className":1838},[748],[579,1840],{"className":1841,"style":744},[743],[579,1843,1845,1848,1851,1854,1857],{"className":1844},[677],[579,1846],{"className":1847,"style":1018},[681],[579,1849,1324],{"className":1850},[686],[579,1852],{"className":1853,"style":744},[743],[579,1855,616],{"className":1856},[748],[579,1858],{"className":1859,"style":744},[743],[579,1861,1863,1866,1869,1872,1875],{"className":1862},[677],[579,1864],{"className":1865,"style":1018},[681],[579,1867,1367],{"className":1868},[686],[579,1870],{"className":1871,"style":798},[743],[579,1873,626],{"className":1874},[802],[579,1876],{"className":1877,"style":798},[743],[579,1879,1881,1884,1887,1890,1893],{"className":1880},[677],[579,1882],{"className":1883,"style":1784},[681],[579,1885,1344],{"className":1886},[686],[579,1888],{"className":1889,"style":744},[743],[579,1891,1355],{"className":1892},[748],[579,1894],{"className":1895,"style":744},[743],[579,1897,1899,1902,1905,1908,1911],{"className":1898},[677],[579,1900],{"className":1901,"style":1784},[681],[579,1903,1409],{"className":1904},[686],[579,1906],{"className":1907,"style":744},[743],[579,1909,1380],{"className":1910},[748],[579,1912],{"className":1913,"style":744},[743],[579,1915,1917,1920,1923,1926,1929],{"className":1916},[677],[579,1918],{"className":1919,"style":1018},[681],[579,1921,1414],{"className":1922},[686],[579,1924],{"className":1925,"style":798},[743],[579,1927,626],{"className":1928},[802],[579,1930],{"className":1931,"style":798},[743],[579,1933,1935,1938],{"className":1934},[677],[579,1936],{"className":1937,"style":1018},[681],[579,1939,1419],{"className":1940},[686],[564,1942,1943,1944,2084,2085,2297,2298,1261],{},"Ein Vektor mit sich selbst: Für ",[579,1945,1947,1989],{"className":1946},[586],[579,1948,1950],{"className":1949},[590],[592,1951,1952],{"xmlns":594},[597,1953,1954,1986],{},[600,1955,1956,1962,1964,1966,1968,1970,1974,1976,1978,1980,1982,1984],{},[603,1957,1958,1960],{"accent":605},[607,1959,609],{},[611,1961,613],{},[611,1963,626],{},[611,1965,1288],{"stretchy":1304},[1004,1967,1072],{},[611,1969,664],{"separator":605},[1971,1972,1973],"mtext",{}," ",[611,1975,1355],{},[1004,1977,1307],{},[611,1979,664],{"separator":605},[1971,1981,1973],{},[1004,1983,1072],{},[611,1985,1327],{"stretchy":1304},[666,1987,1988],{"encoding":668},"\\vec{a} = (2,\\ -1,\\ 2)",[579,1990,1992,2042],{"className":1991,"ariaHidden":605},[673],[579,1993,1995,1998,2033,2036,2039],{"className":1994},[677],[579,1996],{"className":1997,"style":682},[681],[579,1999,2001],{"className":2000},[686,687],[579,2002,2004],{"className":2003},[691],[579,2005,2007],{"className":2006},[695],[579,2008,2010,2018],{"className":2009,"style":682},[699],[579,2011,2012,2015],{"style":702},[579,2013],{"className":2014,"style":707},[706],[579,2016,609],{"className":2017},[686,711],[579,2019,2020,2023],{"style":702},[579,2021],{"className":2022,"style":707},[706],[579,2024,2026],{"className":2025,"style":721},[720],[579,2027,2029],{"className":2028,"style":726},[725],[728,2030,2031],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,2032],{"d":739},[579,2034],{"className":2035,"style":798},[743],[579,2037,626],{"className":2038},[802],[579,2040],{"className":2041,"style":798},[743],[579,2043,2045,2048,2051,2054,2057,2060,2063,2066,2069,2072,2075,2078,2081],{"className":2044},[677],[579,2046],{"className":2047,"style":812},[681],[579,2049,1288],{"className":2050},[1440],[579,2052,1072],{"className":2053},[686],[579,2055,664],{"className":2056},[941],[579,2058,1973],{"className":2059},[743],[579,2061],{"className":2062,"style":934},[743],[579,2064,1355],{"className":2065},[686],[579,2067,1307],{"className":2068},[686],[579,2070,664],{"className":2071},[941],[579,2073,1973],{"className":2074},[743],[579,2076],{"className":2077,"style":934},[743],[579,2079,1072],{"className":2080},[686],[579,2082,1327],{"className":2083},[1558]," ist ",[579,2086,2088,2131],{"className":2087},[586],[579,2089,2091],{"className":2090},[590],[592,2092,2093],{"xmlns":594},[597,2094,2095,2128],{},[600,2096,2097,2103,2105,2111,2113,2115,2117,2119,2121,2123,2125],{},[603,2098,2099,2101],{"accent":605},[607,2100,609],{},[611,2102,613],{},[611,2104,616],{},[603,2106,2107,2109],{"accent":605},[607,2108,609],{},[611,2110,613],{},[611,2112,626],{},[1004,2114,1344],{},[611,2116,1380],{},[1004,2118,1307],{},[611,2120,1380],{},[1004,2122,1344],{},[611,2124,626],{},[1004,2126,2127],{},"9",[666,2129,2130],{"encoding":668},"\\vec{a} \\cdot \\vec{a} = 4 + 1 + 4 = 9",[579,2132,2134,2184,2234,2252,2270,2288],{"className":2133,"ariaHidden":605},[673],[579,2135,2137,2140,2175,2178,2181],{"className":2136},[677],[579,2138],{"className":2139,"style":682},[681],[579,2141,2143],{"className":2142},[686,687],[579,2144,2146],{"className":2145},[691],[579,2147,2149],{"className":2148},[695],[579,2150,2152,2160],{"className":2151,"style":682},[699],[579,2153,2154,2157],{"style":702},[579,2155],{"className":2156,"style":707},[706],[579,2158,609],{"className":2159},[686,711],[579,2161,2162,2165],{"style":702},[579,2163],{"className":2164,"style":707},[706],[579,2166,2168],{"className":2167,"style":721},[720],[579,2169,2171],{"className":2170,"style":726},[725],[728,2172,2173],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,2174],{"d":739},[579,2176],{"className":2177,"style":744},[743],[579,2179,616],{"className":2180},[748],[579,2182],{"className":2183,"style":744},[743],[579,2185,2187,2190,2225,2228,2231],{"className":2186},[677],[579,2188],{"className":2189,"style":682},[681],[579,2191,2193],{"className":2192},[686,687],[579,2194,2196],{"className":2195},[691],[579,2197,2199],{"className":2198},[695],[579,2200,2202,2210],{"className":2201,"style":682},[699],[579,2203,2204,2207],{"style":702},[579,2205],{"className":2206,"style":707},[706],[579,2208,609],{"className":2209},[686,711],[579,2211,2212,2215],{"style":702},[579,2213],{"className":2214,"style":707},[706],[579,2216,2218],{"className":2217,"style":721},[720],[579,2219,2221],{"className":2220,"style":726},[725],[728,2222,2223],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,2224],{"d":739},[579,2226],{"className":2227,"style":798},[743],[579,2229,626],{"className":2230},[802],[579,2232],{"className":2233,"style":798},[743],[579,2235,2237,2240,2243,2246,2249],{"className":2236},[677],[579,2238],{"className":2239,"style":1784},[681],[579,2241,1344],{"className":2242},[686],[579,2244],{"className":2245,"style":744},[743],[579,2247,1380],{"className":2248},[748],[579,2250],{"className":2251,"style":744},[743],[579,2253,2255,2258,2261,2264,2267],{"className":2254},[677],[579,2256],{"className":2257,"style":1784},[681],[579,2259,1307],{"className":2260},[686],[579,2262],{"className":2263,"style":744},[743],[579,2265,1380],{"className":2266},[748],[579,2268],{"className":2269,"style":744},[743],[579,2271,2273,2276,2279,2282,2285],{"className":2272},[677],[579,2274],{"className":2275,"style":1018},[681],[579,2277,1344],{"className":2278},[686],[579,2280],{"className":2281,"style":798},[743],[579,2283,626],{"className":2284},[802],[579,2286],{"className":2287,"style":798},[743],[579,2289,2291,2294],{"className":2290},[677],[579,2292],{"className":2293,"style":1018},[681],[579,2295,2127],{"className":2296},[686],", also ",[579,2299,2301,2334],{"className":2300},[586],[579,2302,2304],{"className":2303},[590],[592,2305,2306],{"xmlns":594},[597,2307,2308,2331],{},[600,2309,2310,2312,2318,2320,2322,2327,2329],{},[607,2311,630],{"mathvariant":629},[603,2313,2314,2316],{"accent":605},[607,2315,609],{},[611,2317,613],{},[607,2319,630],{"mathvariant":629},[611,2321,626],{},[2323,2324,2325],"msqrt",{},[1004,2326,2127],{},[611,2328,626],{},[1004,2330,1324],{},[666,2332,2333],{"encoding":668},"|\\vec{a}| = \\sqrt{9} = 3",[579,2335,2337,2393,2469],{"className":2336,"ariaHidden":605},[673],[579,2338,2340,2343,2346,2381,2384,2387,2390],{"className":2339},[677],[579,2341],{"className":2342,"style":812},[681],[579,2344,630],{"className":2345},[686],[579,2347,2349],{"className":2348},[686,687],[579,2350,2352],{"className":2351},[691],[579,2353,2355],{"className":2354},[695],[579,2356,2358,2366],{"className":2357,"style":682},[699],[579,2359,2360,2363],{"style":702},[579,2361],{"className":2362,"style":707},[706],[579,2364,609],{"className":2365},[686,711],[579,2367,2368,2371],{"style":702},[579,2369],{"className":2370,"style":707},[706],[579,2372,2374],{"className":2373,"style":721},[720],[579,2375,2377],{"className":2376,"style":726},[725],[728,2378,2379],{"xmlns":730,"width":731,"height":732,"style":733,"viewBox":734,"preserveAspectRatio":735},[737,2380],{"d":739},[579,2382,630],{"className":2383},[686],[579,2385],{"className":2386,"style":798},[743],[579,2388,626],{"className":2389},[802],[579,2391],{"className":2392,"style":798},[743],[579,2394,2396,2400,2460,2463,2466],{"className":2395},[677],[579,2397],{"className":2398,"style":2399},[681],"height:1.04em;vertical-align:-0.1328em;",[579,2401,2404],{"className":2402},[686,2403],"sqrt",[579,2405,2407,2451],{"className":2406},[691,1449],[579,2408,2410,2448],{"className":2409},[695],[579,2411,2414,2428],{"className":2412,"style":2413},[699],"height:0.9072em;",[579,2415,2418,2421],{"className":2416,"style":702},[2417],"svg-align",[579,2419],{"className":2420,"style":707},[706],[579,2422,2425],{"className":2423,"style":2424},[686],"padding-left:0.833em;",[579,2426,2127],{"className":2427},[686],[579,2429,2431,2434],{"style":2430},"top:-2.8672em;",[579,2432],{"className":2433,"style":707},[706],[579,2435,2439],{"className":2436,"style":2438},[2437],"hide-tail","min-width:0.853em;height:1.08em;",[728,2440,2445],{"xmlns":730,"width":2441,"height":2442,"viewBox":2443,"preserveAspectRatio":2444},"400em","1.08em","0 0 400000 1080","xMinYMin slice",[737,2446],{"d":2447},"M95,702\nc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14\nc0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54\nc44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10\ns173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429\nc69,-144,104.5,-217.7,106.5,-221\nl0 -0\nc5.3,-9.3,12,-14,20,-14\nH400000v40H845.2724\ns-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7\nc-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47z\nM834 80h400000v40h-400000z",[579,2449,1479],{"className":2450},[1478],[579,2452,2454],{"className":2453},[695],[579,2455,2458],{"className":2456,"style":2457},[699],"height:0.1328em;",[579,2459],{},[579,2461],{"className":2462,"style":798},[743],[579,2464,626],{"className":2465},[802],[579,2467],{"className":2468,"style":798},[743],[579,2470,2472,2475],{"className":2471},[677],[579,2473],{"className":2474,"style":1018},[681],[579,2476,1324],{"className":2477},[686],[564,2479,2480,2481,2787],{},"In der Ebene: ",[579,2482,2484,2564],{"className":2483},[586],[579,2485,2487],{"className":2486},[590],[592,2488,2489],{"xmlns":594},[597,2490,2491,2561],{},[600,2492,2493,2517,2519,2547,2549,2551,2553,2555,2557,2559],{},[600,2494,2495,2497,2515],{},[611,2496,1288],{"fence":605},[1290,2498,2499,2507],{"rowspacing":1292,"columnalign":1293,"columnspacing":1294},[1296,2500,2501],{},[1299,2502,2503],{},[1302,2504,2505],{"scriptlevel":1006,"displaystyle":1304},[1004,2506,1324],{},[1296,2508,2509],{},[1299,2510,2511],{},[1302,2512,2513],{"scriptlevel":1006,"displaystyle":1304},[1004,2514,1344],{},[611,2516,1327],{"fence":605},[611,2518,616],{},[600,2520,2521,2523,2545],{},[611,2522,1288],{"fence":605},[1290,2524,2525,2537],{"rowspacing":1292,"columnalign":1293,"columnspacing":1294},[1296,2526,2527],{},[1299,2528,2529],{},[1302,2530,2531],{"scriptlevel":1006,"displaystyle":1304},[600,2532,2533,2535],{},[611,2534,1355],{},[1004,2536,1344],{},[1296,2538,2539],{},[1299,2540,2541],{},[1302,2542,2543],{"scriptlevel":1006,"displaystyle":1304},[1004,2544,1324],{},[611,2546,1327],{"fence":605},[611,2548,626],{},[611,2550,1355],{},[1004,2552,1419],{},[611,2554,1380],{},[1004,2556,1419],{},[611,2558,626],{},[1004,2560,1006],{},[666,2562,2563],{"encoding":668},"\\begin{pmatrix} 3 \\\\ 4 \\end{pmatrix} \\cdot \\begin{pmatrix} -4 \\\\ 3 \\end{pmatrix} = -12 + 12 = 0",[579,2565,2567,2655,2739,2760,2778],{"className":2566,"ariaHidden":605},[673],[579,2568,2570,2574,2646,2649,2652],{"className":2569},[677],[579,2571],{"className":2572,"style":2573},[681],"height:2.4em;vertical-align:-0.95em;",[579,2575,2577,2585,2640],{"className":2576},[1436],[579,2578,2582],{"className":2579,"style":2581},[1440,2580],"delimcenter","top:0em;",[579,2583,1288],{"className":2584},[1444,1256],[579,2586,2588],{"className":2587},[686],[579,2589,2591],{"className":2590},[1290],[579,2592,2594],{"className":2593},[1498],[579,2595,2597,2631],{"className":2596},[691,1449],[579,2598,2600,2628],{"className":2599},[695],[579,2601,2604,2616],{"className":2602,"style":2603},[699],"height:1.45em;",[579,2605,2607,2610],{"style":2606},"top:-3.61em;",[579,2608],{"className":2609,"style":707},[706],[579,2611,2613],{"className":2612},[686],[579,2614,1324],{"className":2615},[686],[579,2617,2619,2622],{"style":2618},"top:-2.41em;",[579,2620],{"className":2621,"style":707},[706],[579,2623,2625],{"className":2624},[686],[579,2626,1344],{"className":2627},[686],[579,2629,1479],{"className":2630},[1478],[579,2632,2634],{"className":2633},[695],[579,2635,2638],{"className":2636,"style":2637},[699],"height:0.95em;",[579,2639],{},[579,2641,2643],{"className":2642,"style":2581},[1558,2580],[579,2644,1327],{"className":2645},[1444,1256],[579,2647],{"className":2648,"style":744},[743],[579,2650,616],{"className":2651},[748],[579,2653],{"className":2654,"style":744},[743],[579,2656,2658,2661,2730,2733,2736],{"className":2657},[677],[579,2659],{"className":2660,"style":2573},[681],[579,2662,2664,2670,2724],{"className":2663},[1436],[579,2665,2667],{"className":2666,"style":2581},[1440,2580],[579,2668,1288],{"className":2669},[1444,1256],[579,2671,2673],{"className":2672},[686],[579,2674,2676],{"className":2675},[1290],[579,2677,2679],{"className":2678},[1498],[579,2680,2682,2716],{"className":2681},[691,1449],[579,2683,2685,2713],{"className":2684},[695],[579,2686,2688,2702],{"className":2687,"style":2603},[699],[579,2689,2690,2693],{"style":2606},[579,2691],{"className":2692,"style":707},[706],[579,2694,2696,2699],{"className":2695},[686],[579,2697,1355],{"className":2698},[686],[579,2700,1344],{"className":2701},[686],[579,2703,2704,2707],{"style":2618},[579,2705],{"className":2706,"style":707},[706],[579,2708,2710],{"className":2709},[686],[579,2711,1324],{"className":2712},[686],[579,2714,1479],{"className":2715},[1478],[579,2717,2719],{"className":2718},[695],[579,2720,2722],{"className":2721,"style":2637},[699],[579,2723],{},[579,2725,2727],{"className":2726,"style":2581},[1558,2580],[579,2728,1327],{"className":2729},[1444,1256],[579,2731],{"className":2732,"style":798},[743],[579,2734,626],{"className":2735},[802],[579,2737],{"className":2738,"style":798},[743],[579,2740,2742,2745,2748,2751,2754,2757],{"className":2741},[677],[579,2743],{"className":2744,"style":1784},[681],[579,2746,1355],{"className":2747},[686],[579,2749,1419],{"className":2750},[686],[579,2752],{"className":2753,"style":744},[743],[579,2755,1380],{"className":2756},[748],[579,2758],{"className":2759,"style":744},[743],[579,2761,2763,2766,2769,2772,2775],{"className":2762},[677],[579,2764],{"className":2765,"style":1018},[681],[579,2767,1419],{"className":2768},[686],[579,2770],{"className":2771,"style":798},[743],[579,2773,626],{"className":2774},[802],[579,2776],{"className":2777,"style":798},[743],[579,2779,2781,2784],{"className":2780},[677],[579,2782],{"className":2783,"style":1018},[681],[579,2785,1006],{"className":2786},[686],". Die beiden Vektoren stehen senkrecht aufeinander.",[559,2789,2791],{"id":2790},"typische-fehler","Typische Fehler",[976,2793,2794,2878,2881],{},[979,2795,2796,2797,2877],{},"Als Ergebnis einen Vektor angeben. Das Skalarprodukt ist eine Zahl; der Vektor ",[579,2798,2800,2832],{"className":2799},[586],[579,2801,2803],{"className":2802},[590],[592,2804,2805],{"xmlns":594},[597,2806,2807,2829],{},[600,2808,2809,2811,2813,2815,2817,2819,2821,2823,2825,2827],{},[611,2810,1288],{"stretchy":1304},[1004,2812,1344],{},[611,2814,664],{"separator":605},[1971,2816,1973],{},[611,2818,1355],{},[1004,2820,1409],{},[611,2822,664],{"separator":605},[1971,2824,1973],{},[1004,2826,1414],{},[611,2828,1327],{"stretchy":1304},[666,2830,2831],{"encoding":668},"(4,\\ -10,\\ 18)",[579,2833,2835],{"className":2834,"ariaHidden":605},[673],[579,2836,2838,2841,2844,2847,2850,2853,2856,2859,2862,2865,2868,2871,2874],{"className":2837},[677],[579,2839],{"className":2840,"style":812},[681],[579,2842,1288],{"className":2843},[1440],[579,2845,1344],{"className":2846},[686],[579,2848,664],{"className":2849},[941],[579,2851,1973],{"className":2852},[743],[579,2854],{"className":2855,"style":934},[743],[579,2857,1355],{"className":2858},[686],[579,2860,1409],{"className":2861},[686],[579,2863,664],{"className":2864},[941],[579,2866,1973],{"className":2867},[743],[579,2869],{"className":2870,"style":934},[743],[579,2872,1414],{"className":2873},[686],[579,2875,1327],{"className":2876},[1558]," ist kein Ergebnis.",[979,2879,2880],{},"Skalarprodukt und Kreuzprodukt verwechseln. Das Kreuzprodukt liefert einen Vektor und gibt es nur im Raum.",[979,2882,2883,2884,2977,2978,1261],{},"Vorzeichen vergessen: ",[579,2885,2887,2917],{"className":2886},[586],[579,2888,2890],{"className":2889},[590],[592,2891,2892],{"xmlns":594},[597,2893,2894,2914],{},[600,2895,2896,2898,2900,2902,2904,2906,2908,2910,2912],{},[1004,2897,1072],{},[611,2899,616],{},[611,2901,1288],{"stretchy":1304},[611,2903,1355],{},[1004,2905,1358],{},[611,2907,1327],{"stretchy":1304},[611,2909,626],{},[611,2911,1355],{},[1004,2913,1409],{},[666,2915,2916],{"encoding":668},"2 \\cdot (-5) = -10",[579,2918,2920,2938,2965],{"className":2919,"ariaHidden":605},[673],[579,2921,2923,2926,2929,2932,2935],{"className":2922},[677],[579,2924],{"className":2925,"style":1018},[681],[579,2927,1072],{"className":2928},[686],[579,2930],{"className":2931,"style":744},[743],[579,2933,616],{"className":2934},[748],[579,2936],{"className":2937,"style":744},[743],[579,2939,2941,2944,2947,2950,2953,2956,2959,2962],{"className":2940},[677],[579,2942],{"className":2943,"style":812},[681],[579,2945,1288],{"className":2946},[1440],[579,2948,1355],{"className":2949},[686],[579,2951,1358],{"className":2952},[686],[579,2954,1327],{"className":2955},[1558],[579,2957],{"className":2958,"style":798},[743],[579,2960,626],{"className":2961},[802],[579,2963],{"className":2964,"style":798},[743],[579,2966,2968,2971,2974],{"className":2967},[677],[579,2969],{"className":2970,"style":1784},[681],[579,2972,1355],{"className":2973},[686],[579,2975,1409],{"className":2976},[686],", nicht ",[579,2979,2981,2994],{"className":2980},[586],[579,2982,2984],{"className":2983},[590],[592,2985,2986],{"xmlns":594},[597,2987,2988,2992],{},[600,2989,2990],{},[1004,2991,1409],{},[666,2993,1409],{"encoding":668},[579,2995,2997],{"className":2996,"ariaHidden":605},[673],[579,2998,3000,3003],{"className":2999},[677],[579,3001],{"className":3002,"style":1018},[681],[579,3004,1409],{"className":3005},[686],{"title":3007,"searchDepth":3008,"depth":3008,"links":3009},"",2,[3010,3011,3012],{"id":561,"depth":3008,"text":562},{"id":1267,"depth":3008,"text":1268},{"id":2790,"depth":3008,"text":2791},[3014,3017,3019,3022,3024,3026,3028],{"lhs":3015,"rhs":3016},"a1*b1 + a2*b2 + a3*b3","dot([a1, a2, a3], [b1, b2, b3])",{"lhs":3015,"rhs":3018},"b1*a1 + b2*a2 + b3*a3",{"lhs":3020,"rhs":3021},"a1*a1 + a2*a2 + a3*a3","sqrt(a1^2 + a2^2 + a3^2)^2",{"lhs":3023,"rhs":1419},"1*4 + 2*(-5) + 3*6",{"lhs":3025,"rhs":2127},"2*2 + (-1)*(-1) + 2*2",{"lhs":3027,"rhs":1324},"sqrt(9)",{"lhs":3029,"rhs":1006},"3*(-4) + 4*3",null,"md","dot-product","\\vec{a} \\cdot \\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3","\u003Cspan class=\"katex-display\">\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\" display=\"block\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>a\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>b\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>=\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmn>1\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmn>2\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmo>+\u003C\u002Fmo>\u003Cmsub>\u003Cmi>a\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003Cmsub>\u003Cmi>b\u003C\u002Fmi>\u003Cmn>3\u003C\u002Fmn>\u003C\u002Fmsub>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\vec{a} \\cdot \\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3\u003C\u002Fannotation>\u003C\u002Fsemantics>\u003C\u002Fmath>\u003C\u002Fspan>\u003Cspan class=\"katex-html\" aria-hidden=\"true\">\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" 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