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",[568,1113,1115,1134],{"className":1114},[571],[568,1116,1118],{"className":1117},[575],[577,1119,1120],{"xmlns":579},[581,1121,1122,1131],{},[584,1123,1124],{},[973,1125,1126,1129],{},[987,1127,1128],{},"180",[595,1130,1068],{},[647,1132,1133],{"encoding":649},"180^\\circ",[568,1135,1137],{"className":1136,"ariaHidden":589},[654],[568,1138,1140,1143,1147],{"className":1139},[658],[568,1141],{"className":1142,"style":1081},[662],[568,1144,1146],{"className":1145},[667],"18",[568,1148,1150,1153],{"className":1149},[667],[568,1151,1065],{"className":1152},[667],[568,1154,1156],{"className":1155},[1012],[568,1157,1159],{"className":1158},[672],[568,1160,1162],{"className":1161},[676],[568,1163,1165],{"className":1164,"style":1081},[680],[568,1166,1167,1170],{"style":1024},[568,1168],{"className":1169,"style":1028},[687],[568,1171,1173],{"className":1172},[1032,1033,1034,1035],[568,1174,1068],{"className":1175},[729,1035],":",[1178,1179,1180,1332,1480],"ul",{},[1181,1182,1183,1331],"li",{},[568,1184,1186,1217],{"className":1185},[571],[568,1187,1189],{"className":1188},[575],[577,1190,1191],{"xmlns":579},[581,1192,1193,1214],{},[584,1194,1195,1201,1203,1209,1212],{},[587,1196,1197,1199],{"accent":589},[591,1198,593],{},[595,1200,597],{},[595,1202,600],{},[587,1204,1205,1207],{"accent":589},[591,1206,605],{},[595,1208,597],{},[595,1210,1211],{},">",[987,1213,1065],{},[647,1215,1216],{"encoding":649},"\\vec{a} \\cdot \\vec{b} > 0",[568,1218,1220,1270,1321],{"className":1219,"ariaHidden":589},[654],[568,1221,1223,1226,1261,1264,1267],{"className":1222},[658],[568,1224],{"className":1225,"style":663},[662],[568,1227,1229],{"className":1228},[667,668],[568,1230,1232],{"className":1231},[672],[568,1233,1235],{"className":1234},[676],[568,1236,1238,1246],{"className":1237,"style":663},[680],[568,1239,1240,1243],{"style":683},[568,1241],{"className":1242,"style":688},[687],[568,1244,593],{"className":1245},[667,692],[568,1247,1248,1251],{"style":683},[568,1249],{"className":1250,"style":688},[687],[568,1252,1254],{"className":1253,"style":702},[701],[568,1255,1257],{"className":1256,"style":707},[706],[709,1258,1259],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1260],{"d":720},[568,1262],{"className":1263,"style":725},[724],[568,1265,600],{"className":1266},[729],[568,1268],{"className":1269,"style":725},[724],[568,1271,1273,1277,1312,1315,1318],{"className":1272},[658],[568,1274],{"className":1275,"style":1276},[662],"height:1.0165em;vertical-align:-0.0391em;",[568,1278,1280],{"className":1279},[667,668],[568,1281,1283],{"className":1282},[672],[568,1284,1286],{"className":1285},[676],[568,1287,1289,1297],{"className":1288,"style":739},[680],[568,1290,1291,1294],{"style":683},[568,1292],{"className":1293,"style":688},[687],[568,1295,605],{"className":1296},[667,692],[568,1298,1299,1302],{"style":762},[568,1300],{"className":1301,"style":688},[687],[568,1303,1305],{"className":1304,"style":702},[701],[568,1306,1308],{"className":1307,"style":707},[706],[709,1309,1310],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1311],{"d":720},[568,1313],{"className":1314,"style":779},[724],[568,1316,1211],{"className":1317},[783],[568,1319],{"className":1320,"style":779},[724],[568,1322,1324,1328],{"className":1323},[658],[568,1325],{"className":1326,"style":1327},[662],"height:0.6444em;",[568,1329,1065],{"className":1330},[667],": spitzer Winkel,",[1181,1333,1334,1479],{},[568,1335,1337,1367],{"className":1336},[571],[568,1338,1340],{"className":1339},[575],[577,1341,1342],{"xmlns":579},[581,1343,1344,1364],{},[584,1345,1346,1352,1354,1360,1362],{},[587,1347,1348,1350],{"accent":589},[591,1349,593],{},[595,1351,597],{},[595,1353,600],{},[587,1355,1356,1358],{"accent":589},[591,1357,605],{},[595,1359,597],{},[595,1361,610],{},[987,1363,1065],{},[647,1365,1366],{"encoding":649},"\\vec{a} \\cdot \\vec{b} = 0",[568,1368,1370,1420,1470],{"className":1369,"ariaHidden":589},[654],[568,1371,1373,1376,1411,1414,1417],{"className":1372},[658],[568,1374],{"className":1375,"style":663},[662],[568,1377,1379],{"className":1378},[667,668],[568,1380,1382],{"className":1381},[672],[568,1383,1385],{"className":1384},[676],[568,1386,1388,1396],{"className":1387,"style":663},[680],[568,1389,1390,1393],{"style":683},[568,1391],{"className":1392,"style":688},[687],[568,1394,593],{"className":1395},[667,692],[568,1397,1398,1401],{"style":683},[568,1399],{"className":1400,"style":688},[687],[568,1402,1404],{"className":1403,"style":702},[701],[568,1405,1407],{"className":1406,"style":707},[706],[709,1408,1409],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1410],{"d":720},[568,1412],{"className":1413,"style":725},[724],[568,1415,600],{"className":1416},[729],[568,1418],{"className":1419,"style":725},[724],[568,1421,1423,1426,1461,1464,1467],{"className":1422},[658],[568,1424],{"className":1425,"style":739},[662],[568,1427,1429],{"className":1428},[667,668],[568,1430,1432],{"className":1431},[672],[568,1433,1435],{"className":1434},[676],[568,1436,1438,1446],{"className":1437,"style":739},[680],[568,1439,1440,1443],{"style":683},[568,1441],{"className":1442,"style":688},[687],[568,1444,605],{"className":1445},[667,692],[568,1447,1448,1451],{"style":762},[568,1449],{"className":1450,"style":688},[687],[568,1452,1454],{"className":1453,"style":702},[701],[568,1455,1457],{"className":1456,"style":707},[706],[709,1458,1459],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1460],{"d":720},[568,1462],{"className":1463,"style":779},[724],[568,1465,610],{"className":1466},[783],[568,1468],{"className":1469,"style":779},[724],[568,1471,1473,1476],{"className":1472},[658],[568,1474],{"className":1475,"style":1327},[662],[568,1477,1065],{"className":1478},[667],": rechter Winkel,",[1181,1481,1482,1628],{},[568,1483,1485,1516],{"className":1484},[571],[568,1486,1488],{"className":1487},[575],[577,1489,1490],{"xmlns":579},[581,1491,1492,1513],{},[584,1493,1494,1500,1502,1508,1511],{},[587,1495,1496,1498],{"accent":589},[591,1497,593],{},[595,1499,597],{},[595,1501,600],{},[587,1503,1504,1506],{"accent":589},[591,1505,605],{},[595,1507,597],{},[595,1509,1510],{},"\u003C",[987,1512,1065],{},[647,1514,1515],{"encoding":649},"\\vec{a} \\cdot \\vec{b} \u003C 0",[568,1517,1519,1569,1619],{"className":1518,"ariaHidden":589},[654],[568,1520,1522,1525,1560,1563,1566],{"className":1521},[658],[568,1523],{"className":1524,"style":663},[662],[568,1526,1528],{"className":1527},[667,668],[568,1529,1531],{"className":1530},[672],[568,1532,1534],{"className":1533},[676],[568,1535,1537,1545],{"className":1536,"style":663},[680],[568,1538,1539,1542],{"style":683},[568,1540],{"className":1541,"style":688},[687],[568,1543,593],{"className":1544},[667,692],[568,1546,1547,1550],{"style":683},[568,1548],{"className":1549,"style":688},[687],[568,1551,1553],{"className":1552,"style":702},[701],[568,1554,1556],{"className":1555,"style":707},[706],[709,1557,1558],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1559],{"d":720},[568,1561],{"className":1562,"style":725},[724],[568,1564,600],{"className":1565},[729],[568,1567],{"className":1568,"style":725},[724],[568,1570,1572,1575,1610,1613,1616],{"className":1571},[658],[568,1573],{"className":1574,"style":1276},[662],[568,1576,1578],{"className":1577},[667,668],[568,1579,1581],{"className":1580},[672],[568,1582,1584],{"className":1583},[676],[568,1585,1587,1595],{"className":1586,"style":739},[680],[568,1588,1589,1592],{"style":683},[568,1590],{"className":1591,"style":688},[687],[568,1593,605],{"className":1594},[667,692],[568,1596,1597,1600],{"style":762},[568,1598],{"className":1599,"style":688},[687],[568,1601,1603],{"className":1602,"style":702},[701],[568,1604,1606],{"className":1605,"style":707},[706],[709,1607,1608],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1609],{"d":720},[568,1611],{"className":1612,"style":779},[724],[568,1614,1510],{"className":1615},[783],[568,1617],{"className":1618,"style":779},[724],[568,1620,1622,1625],{"className":1621},[658],[568,1623],{"className":1624,"style":1327},[662],[568,1626,1065],{"className":1627},[667],": stumpfer Winkel.",[564,1630,1631,1632,1660],{},"Für den Nullvektor ist kein Winkel definiert, weil man durch ",[568,1633,1635,1648],{"className":1634},[571],[568,1636,1638],{"className":1637},[575],[577,1639,1640],{"xmlns":579},[581,1641,1642,1646],{},[584,1643,1644],{},[987,1645,1065],{},[647,1647,1065],{"encoding":649},[568,1649,1651],{"className":1650,"ariaHidden":589},[654],[568,1652,1654,1657],{"className":1653},[658],[568,1655],{"className":1656,"style":1327},[662],[568,1658,1065],{"className":1659},[667]," teilen müsste.",[559,1662,1664],{"id":1663},"beispiele","Beispiele",[564,1666,1667,1668,1111,1810,1176],{},"Für ",[568,1669,1671,1715],{"className":1670},[571],[568,1672,1674],{"className":1673},[575],[577,1675,1676],{"xmlns":579},[581,1677,1678,1712],{},[584,1679,1680,1686,1688,1692,1694,1697,1701,1703,1705,1707,1709],{},[587,1681,1682,1684],{"accent":589},[591,1683,593],{},[595,1685,597],{},[595,1687,610],{},[595,1689,1691],{"stretchy":1690},"false","(",[987,1693,989],{},[595,1695,1696],{"separator":589},",",[1698,1699,1700],"mtext",{}," ",[987,1702,1065],{},[595,1704,1696],{"separator":589},[1698,1706,1700],{},[987,1708,989],{},[595,1710,1711],{"stretchy":1690},")",[647,1713,1714],{"encoding":649},"\\vec{a} = (1,\\ 0,\\ 1)",[568,1716,1718,1768],{"className":1717,"ariaHidden":589},[654],[568,1719,1721,1724,1759,1762,1765],{"className":1720},[658],[568,1722],{"className":1723,"style":663},[662],[568,1725,1727],{"className":1726},[667,668],[568,1728,1730],{"className":1729},[672],[568,1731,1733],{"className":1732},[676],[568,1734,1736,1744],{"className":1735,"style":663},[680],[568,1737,1738,1741],{"style":683},[568,1739],{"className":1740,"style":688},[687],[568,1742,593],{"className":1743},[667,692],[568,1745,1746,1749],{"style":683},[568,1747],{"className":1748,"style":688},[687],[568,1750,1752],{"className":1751,"style":702},[701],[568,1753,1755],{"className":1754,"style":707},[706],[709,1756,1757],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1758],{"d":720},[568,1760],{"className":1761,"style":779},[724],[568,1763,610],{"className":1764},[783],[568,1766],{"className":1767,"style":779},[724],[568,1769,1771,1774,1778,1781,1785,1788,1791,1794,1797,1800,1803,1806],{"className":1770},[658],[568,1772],{"className":1773,"style":793},[662],[568,1775,1691],{"className":1776},[1777],"mopen",[568,1779,989],{"className":1780},[667],[568,1782,1696],{"className":1783},[1784],"mpunct",[568,1786,1700],{"className":1787},[724],[568,1789],{"className":1790,"style":915},[724],[568,1792,1065],{"className":1793},[667],[568,1795,1696],{"className":1796},[1784],[568,1798,1700],{"className":1799},[724],[568,1801],{"className":1802,"style":915},[724],[568,1804,989],{"className":1805},[667],[568,1807,1711],{"className":1808},[1809],"mclose",[568,1811,1813,1851],{"className":1812},[571],[568,1814,1816],{"className":1815},[575],[577,1817,1818],{"xmlns":579},[581,1819,1820,1848],{},[584,1821,1822,1828,1830,1832,1834,1836,1838,1840,1842,1844,1846],{},[587,1823,1824,1826],{"accent":589},[591,1825,605],{},[595,1827,597],{},[595,1829,610],{},[595,1831,1691],{"stretchy":1690},[987,1833,1065],{},[595,1835,1696],{"separator":589},[1698,1837,1700],{},[987,1839,989],{},[595,1841,1696],{"separator":589},[1698,1843,1700],{},[987,1845,989],{},[595,1847,1711],{"stretchy":1690},[647,1849,1850],{"encoding":649},"\\vec{b} = (0,\\ 1,\\ 1)",[568,1852,1854,1904],{"className":1853,"ariaHidden":589},[654],[568,1855,1857,1860,1895,1898,1901],{"className":1856},[658],[568,1858],{"className":1859,"style":739},[662],[568,1861,1863],{"className":1862},[667,668],[568,1864,1866],{"className":1865},[672],[568,1867,1869],{"className":1868},[676],[568,1870,1872,1880],{"className":1871,"style":739},[680],[568,1873,1874,1877],{"style":683},[568,1875],{"className":1876,"style":688},[687],[568,1878,605],{"className":1879},[667,692],[568,1881,1882,1885],{"style":762},[568,1883],{"className":1884,"style":688},[687],[568,1886,1888],{"className":1887,"style":702},[701],[568,1889,1891],{"className":1890,"style":707},[706],[709,1892,1893],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,1894],{"d":720},[568,1896],{"className":1897,"style":779},[724],[568,1899,610],{"className":1900},[783],[568,1902],{"className":1903,"style":779},[724],[568,1905,1907,1910,1913,1916,1919,1922,1925,1928,1931,1934,1937,1940],{"className":1906},[658],[568,1908],{"className":1909,"style":793},[662],[568,1911,1691],{"className":1912},[1777],[568,1914,1065],{"className":1915},[667],[568,1917,1696],{"className":1918},[1784],[568,1920,1700],{"className":1921},[724],[568,1923],{"className":1924,"style":915},[724],[568,1926,989],{"className":1927},[667],[568,1929,1696],{"className":1930},[1784],[568,1932,1700],{"className":1933},[724],[568,1935],{"className":1936,"style":915},[724],[568,1938,989],{"className":1939},[667],[568,1941,1711],{"className":1942},[1809],[568,1944,1947],{"className":1945},[1946],"katex-display",[568,1948,1950,2028],{"className":1949},[571],[568,1951,1953],{"className":1952},[575],[577,1954,1956],{"xmlns":579,"display":1955},"block",[581,1957,1958,2025],{},[584,1959,1960,1962,1964,1966,1968,1998,2000,2006,2009,2012,2014,2016,2018],{},[591,1961,639],{},[595,1963,642],{},[591,1965,645],{},[595,1967,610],{},[1969,1970,1971,1984],"mfrac",{},[584,1972,1973,1975,1978,1980,1982],{},[987,1974,1065],{},[595,1976,1977],{},"+",[987,1979,1065],{},[595,1981,1977],{},[987,1983,989],{},[584,1985,1986,1992,1994],{},[1987,1988,1989],"msqrt",{},[987,1990,1991],{},"2",[595,1993,600],{},[1987,1995,1996],{},[987,1997,1991],{},[595,1999,610],{},[1969,2001,2002,2004],{},[987,2003,989],{},[987,2005,1991],{},[724,2007],{"width":2008},"1em",[595,2010,2011],{},"⇒",[724,2013],{"width":2008},[591,2015,645],{},[595,2017,610],{},[973,2019,2020,2023],{},[987,2021,2022],{},"60",[595,2024,1068],{},[647,2026,2027],{"encoding":649},"\\cos\\varphi = \\frac{0 + 0 + 1}{\\sqrt{2} \\cdot \\sqrt{2}} = \\frac{1}{2}\n\\quad\\Rightarrow\\quad \\varphi = 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-4",[568,2755,2757,2807,2857,2876,2894,2912],{"className":2756,"ariaHidden":589},[654],[568,2758,2760,2763,2798,2801,2804],{"className":2759},[658],[568,2761],{"className":2762,"style":663},[662],[568,2764,2766],{"className":2765},[667,668],[568,2767,2769],{"className":2768},[672],[568,2770,2772],{"className":2771},[676],[568,2773,2775,2783],{"className":2774,"style":663},[680],[568,2776,2777,2780],{"style":683},[568,2778],{"className":2779,"style":688},[687],[568,2781,593],{"className":2782},[667,692],[568,2784,2785,2788],{"style":683},[568,2786],{"className":2787,"style":688},[687],[568,2789,2791],{"className":2790,"style":702},[701],[568,2792,2794],{"className":2793,"style":707},[706],[709,2795,2796],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,2797],{"d":720},[568,2799],{"className":2800,"style":725},[724],[568,2802,600],{"className":2803},[729],[568,2805],{"className":2806,"style":725},[724],[568,2808,2810,2813,2848,2851,2854],{"className":2809},[658],[568,2811],{"className":2812,"style":739},[662],[568,2814,2816],{"className":2815},[667,668],[568,2817,2819],{"className":2818},[672],[568,2820,2822],{"className":2821},[676],[568,2823,2825,2833],{"className":2824,"style":739},[680],[568,2826,2827,2830],{"style":683},[568,2828],{"className":2829,"style":688},[687],[568,2831,605],{"className":2832},[667,692],[568,2834,2835,2838],{"style":762},[568,2836],{"className":2837,"style":688},[687],[568,2839,2841],{"className":2840,"style":702},[701],[568,2842,2844],{"className":2843,"style":707},[706],[709,2845,2846],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,2847],{"d":720},[568,2849],{"className":2850,"style":779},[724],[568,2852,610],{"className":2853},[783],[568,2855],{"className":2856,"style":779},[724],[568,2858,2860,2864,2867,2870,2873],{"className":2859},[658],[568,2861],{"className":2862,"style":2863},[662],"height:0.7278em;vertical-align:-0.0833em;",[568,2865,1991],{"className":2866},[667],[568,2868],{"className":2869,"style":725},[724],[568,2871,985],{"className":2872},[729],[568,2874],{"className":2875,"style":725},[724],[568,2877,2879,2882,2885,2888,2891],{"className":2878},[658],[568,2880],{"className":2881,"style":2863},[662],[568,2883,1991],{"className":2884},[667],[568,2886],{"className":2887,"style":725},[724],[568,2889,985],{"className":2890},[729],[568,2892],{"className":2893,"style":725},[724],[568,2895,2897,2900,2903,2906,2909],{"className":2896},[658],[568,2898],{"className":2899,"style":1327},[662],[568,2901,2744],{"className":2902},[667],[568,2904],{"className":2905,"style":779},[724],[568,2907,610],{"className":2908},[783],[568,2910],{"className":2911,"style":779},[724],[568,2913,2915,2918,2921],{"className":2914},[658],[568,2916],{"className":2917,"style":2863},[662],[568,2919,985],{"className":2920},[667],[568,2922,2744],{"className":2923},[667],[568,2925,2927,2966],{"className":2926},[571],[568,2928,2930],{"className":2929},[575],[577,2931,2932],{"xmlns":579},[581,2933,2934,2963],{},[584,2935,2936,2938,2944,2946,2948,2950,2956,2958,2960],{},[591,2937,614],{"mathvariant":613},[587,2939,2940,2942],{"accent":589},[591,2941,593],{},[595,2943,597],{},[591,2945,614],{"mathvariant":613},[595,2947,610],{},[591,2949,614],{"mathvariant":613},[587,2951,2952,2954],{"accent":589},[591,2953,605],{},[595,2955,597],{},[591,2957,614],{"mathvariant":613},[595,2959,610],{},[987,2961,2962],{},"3",[647,2964,2965],{"encoding":649},"|\\vec{a}| = |\\vec{b}| = 3",[568,2967,2969,3025,3081],{"className":2968,"ariaHidden":589},[654],[568,2970,2972,2975,2978,3013,3016,3019,3022],{"className":2971},[658],[568,2973],{"className":2974,"style":793},[662],[568,2976,614],{"className":2977},[667],[568,2979,2981],{"className":2980},[667,668],[568,2982,2984],{"className":2983},[672],[568,2985,2987],{"className":2986},[676],[568,2988,2990,2998],{"className":2989,"style":663},[680],[568,2991,2992,2995],{"style":683},[568,2993],{"className":2994,"style":688},[687],[568,2996,593],{"className":2997},[667,692],[568,2999,3000,3003],{"style":683},[568,3001],{"className":3002,"style":688},[687],[568,3004,3006],{"className":3005,"style":702},[701],[568,3007,3009],{"className":3008,"style":707},[706],[709,3010,3011],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,3012],{"d":720},[568,3014,614],{"className":3015},[667],[568,3017],{"className":3018,"style":779},[724],[568,3020,610],{"className":3021},[783],[568,3023],{"className":3024,"style":779},[724],[568,3026,3028,3031,3034,3069,3072,3075,3078],{"className":3027},[658],[568,3029],{"className":3030,"style":850},[662],[568,3032,614],{"className":3033},[667],[568,3035,3037],{"className":3036},[667,668],[568,3038,3040],{"className":3039},[672],[568,3041,3043],{"className":3042},[676],[568,3044,3046,3054],{"className":3045,"style":739},[680],[568,3047,3048,3051],{"style":683},[568,3049],{"className":3050,"style":688},[687],[568,3052,605],{"className":3053},[667,692],[568,3055,3056,3059],{"style":762},[568,3057],{"className":3058,"style":688},[687],[568,3060,3062],{"className":3061,"style":702},[701],[568,3063,3065],{"className":3064,"style":707},[706],[709,3066,3067],{"xmlns":711,"width":712,"height":713,"style":714,"viewBox":715,"preserveAspectRatio":716},[718,3068],{"d":720},[568,3070,614],{"className":3071},[667],[568,3073],{"className":3074,"style":779},[724],[568,3076,610],{"className":3077},[783],[568,3079],{"className":3080,"style":779},[724],[568,3082,3084,3087],{"className":3083},[658],[568,3085],{"className":3086,"style":1327},[662],[568,3088,2962],{"className":3089},[667],[568,3091,3093],{"className":3092},[1946],[568,3094,3096,3161],{"className":3095},[571],[568,3097,3099],{"className":3098},[575],[577,3100,3101],{"xmlns":579,"display":1955},[581,3102,3103,3158],{},[584,3104,3105,3107,3109,3111,3113,3129,3131,3133,3140,3142,3144,3146,3148,3151],{},[591,3106,639],{},[595,3108,642],{},[591,3110,645],{},[595,3112,610],{},[1969,3114,3115,3121],{},[584,3116,3117,3119],{},[595,3118,985],{},[987,3120,2744],{},[584,3122,3123,3125,3127],{},[987,3124,2962],{},[595,3126,600],{},[987,3128,2962],{},[595,3130,610],{},[595,3132,985],{},[1969,3134,3135,3137],{},[987,3136,2744],{},[987,3138,3139],{},"9",[724,3141],{"width":2008},[595,3143,2011],{},[724,3145],{"width":2008},[591,3147,645],{},[595,3149,3150],{},"≈",[973,3152,3153,3156],{},[987,3154,3155],{},"116,39",[595,3157,1068],{},[647,3159,3160],{"encoding":649},"\\cos\\varphi = \\frac{-4}{3 \\cdot 3} = -\\frac{4}{9}\n\\quad\\Rightarrow\\quad \\varphi \\approx 116{,}39^\\circ",[568,3162,3164,3188,3280,3366,3385],{"className":3163,"ariaHidden":589},[654],[568,3165,3167,3170,3173,3176,3179,3182,3185],{"className":3166},[658],[568,3168],{"className":3169,"style":907},[662],[568,3171,639],{"className":3172},[911],[568,3174],{"className":3175,"style":915},[724],[568,3177,645],{"className":3178},[667,692],[568,3180],{"className":3181,"style":779},[724],[568,3183,610],{"className":3184},[783],[568,3186],{"className":3187,"style":779},[724],[568,3189,3191,3194,3271,3274,3277],{"className":3190},[658],[568,3192],{"className":3193,"style":2287},[662],[568,3195,3197,3200,3268],{"className":3196},[667],[568,3198],{"className":3199},[1777,2068],[568,3201,3203],{"className":3202},[1969],[568,3204,3206,3260],{"className":3205},[672,2075],[568,3207,3209,3257],{"className":3208},[676],[568,3210,3212,3235,3243],{"className":3211,"style":2082},[680],[568,3213,3214,3217],{"style":2308},[568,3215],{"className":3216,"style":688},[687],[568,3218,3220,3223,3226,3229,3232],{"className":3219},[667],[568,3221,2962],{"className":3222},[667],[568,3224],{"className":3225,"style":725},[724],[568,3227,600],{"className":3228},[729],[568,3230],{"className":3231,"style":725},[724],[568,3233,2962],{"className":3234},[667],[568,3236,3237,3240],{"style":2212},[568,3238],{"className":3239,"style":688},[687],[568,3241],{"className":3242,"style":2220},[2219],[568,3244,3245,3248],{"style":2223},[568,3246],{"className":3247,"style":688},[687],[568,3249,3251,3254],{"className":3250},[667],[568,3252,985],{"className":3253},[667],[568,3255,2744],{"className":3256},[667],[568,3258,2144],{"className":3259},[2143],[568,3261,3263],{"className":3262},[676],[568,3264,3266],{"className":3265,"style":2346},[680],[568,3267],{},[568,3269],{"className":3270},[1809,2068],[568,3272],{"className":3273,"style":779},[724],[568,3275,610],{"className":3276},[783],[568,3278],{"className":3279,"style":779},[724],[568,3281,3283,3286,3289,3351,3354,3357,3360,3363],{"className":3282},[658],[568,3284],{"className":3285,"style":2287},[662],[568,3287,985],{"className":3288},[667],[568,3290,3292,3295,3348],{"className":3291},[667],[568,3293],{"className":3294},[1777,2068],[568,3296,3298],{"className":3297},[1969],[568,3299,3301,3340],{"className":3300},[672,2075],[568,3302,3304,3337],{"className":3303},[676],[568,3305,3307,3318,3326],{"className":3306,"style":2082},[680],[568,3308,3309,3312],{"style":2308},[568,3310],{"className":3311,"style":688},[687],[568,3313,3315],{"className":3314},[667],[568,3316,3139],{"className":3317},[667],[568,3319,3320,3323],{"style":2212},[568,3321],{"className":3322,"style":688},[687],[568,3324],{"className":3325,"style":2220},[2219],[568,3327,3328,3331],{"style":2223},[568,3329],{"className":3330,"style":688},[687],[568,3332,3334],{"className":3333},[667],[568,3335,2744],{"className":3336},[667],[568,3338,2144],{"className":3339},[2143],[568,3341,3343],{"className":3342},[676],[568,3344,3346],{"className":3345,"style":2346},[680],[568,3347],{},[568,3349],{"className":3350},[1809,2068],[568,3352],{"className":3353,"style":2355},[724],[568,3355],{"className":3356,"style":779},[724],[568,3358,2011],{"className":3359},[783],[568,3361],{"className":3362,"style":2355},[724],[568,3364],{"className":3365,"style":779},[724],[568,3367,3369,3373,3376,3379,3382],{"className":3368},[658],[568,3370],{"className":3371,"style":3372},[662],"height:0.6776em;vertical-align:-0.1944em;",[568,3374,645],{"className":3375},[667,692],[568,3377],{"className":3378,"style":779},[724],[568,3380,3150],{"className":3381},[783],[568,3383],{"className":3384,"style":779},[724],[568,3386,3388,3392,3396,3402,3405],{"className":3387},[658],[568,3389],{"className":3390,"style":3391},[662],"height:0.9186em;vertical-align:-0.1944em;",[568,3393,3395],{"className":3394},[667],"116",[568,3397,3399],{"className":3398},[667],[568,3400,1696],{"className":3401},[1784],[568,3403,2962],{"className":3404},[667],[568,3406,3408,3411],{"className":3407},[667],[568,3409,3139],{"className":3410},[667],[568,3412,3414],{"className":3413},[1012],[568,3415,3417],{"className":3416},[672],[568,3418,3420],{"className":3419},[676],[568,3421,3423],{"className":3422,"style":2392},[680],[568,3424,3425,3428],{"style":2417},[568,3426],{"className":3427,"style":1028},[687],[568,3429,3431],{"className":3430},[1032,1033,1034,1035],[568,3432,1068],{"className":3433},[729,1035],[564,3435,3436],{},"Der Winkel ist stumpf, passend zum negativen Skalarprodukt.",[559,3438,3440],{"id":3439},"typische-fehler","Typische Fehler",[1178,3442,3443,3598,3714],{},[1181,3444,3445,3446,3450,3451,3525,3526,3597],{},"Im Zähler Betragsstriche setzen. Das ist richtig für den Winkel zwischen ",[3447,3448,3449],"strong",{},"Geraden"," oder Ebenen, nicht zwischen Vektoren. Im zweiten Beispiel käme ",[568,3452,3454,3473],{"className":3453},[571],[568,3455,3457],{"className":3456},[575],[577,3458,3459],{"xmlns":579},[581,3460,3461,3470],{},[584,3462,3463],{},[973,3464,3465,3468],{},[987,3466,3467],{},"63,61",[595,3469,1068],{},[647,3471,3472],{"encoding":649},"63{,}61^\\circ",[568,3474,3476],{"className":3475,"ariaHidden":589},[654],[568,3477,3479,3483,3487,3493,3496],{"className":3478},[658],[568,3480],{"className":3481,"style":3482},[662],"height:0.8686em;vertical-align:-0.1944em;",[568,3484,3486],{"className":3485},[667],"63",[568,3488,3490],{"className":3489},[667],[568,3491,1696],{"className":3492},[1784],[568,3494,2396],{"className":3495},[667],[568,3497,3499,3502],{"className":3498},[667],[568,3500,989],{"className":3501},[667],[568,3503,3505],{"className":3504},[1012],[568,3506,3508],{"className":3507},[672],[568,3509,3511],{"className":3510},[676],[568,3512,3514],{"className":3513,"style":1081},[680],[568,3515,3516,3519],{"style":1024},[568,3517],{"className":3518,"style":1028},[687],[568,3520,3522],{"className":3521},[1032,1033,1034,1035],[568,3523,1068],{"className":3524},[729,1035]," heraus, der Nebenwinkel zu ",[568,3527,3529,3547],{"className":3528},[571],[568,3530,3532],{"className":3531},[575],[577,3533,3534],{"xmlns":579},[581,3535,3536,3544],{},[584,3537,3538],{},[973,3539,3540,3542],{},[987,3541,3155],{},[595,3543,1068],{},[647,3545,3546],{"encoding":649},"116{,}39^\\circ",[568,3548,3550],{"className":3549,"ariaHidden":589},[654],[568,3551,3553,3556,3559,3565,3568],{"className":3552},[658],[568,3554],{"className":3555,"style":3482},[662],[568,3557,3395],{"className":3558},[667],[568,3560,3562],{"className":3561},[667],[568,3563,1696],{"className":3564},[1784],[568,3566,2962],{"className":3567},[667],[568,3569,3571,3574],{"className":3570},[667],[568,3572,3139],{"className":3573},[667],[568,3575,3577],{"className":3576},[1012],[568,3578,3580],{"className":3579},[672],[568,3581,3583],{"className":3582},[676],[568,3584,3586],{"className":3585,"style":1081},[680],[568,3587,3588,3591],{"style":1024},[568,3589],{"className":3590,"style":1028},[687],[568,3592,3594],{"className":3593},[1032,1033,1034,1035],[568,3595,1068],{"className":3596},[729,1035],".",[1181,3599,3600,3601,3642,3643,3713],{},"Den Taschenrechner im Bogenmaß lassen und ",[568,3602,3604,3619],{"className":3603},[571],[568,3605,3607],{"className":3606},[575],[577,3608,3609],{"xmlns":579},[581,3610,3611,3616],{},[584,3612,3613],{},[987,3614,3615],{},"2,03",[647,3617,3618],{"encoding":649},"2{,}03",[568,3620,3622],{"className":3621,"ariaHidden":589},[654],[568,3623,3625,3629,3632,3638],{"className":3624},[658],[568,3626],{"className":3627,"style":3628},[662],"height:0.8389em;vertical-align:-0.1944em;",[568,3630,1991],{"className":3631},[667],[568,3633,3635],{"className":3634},[667],[568,3636,1696],{"className":3637},[1784],[568,3639,3641],{"className":3640},[667],"03"," statt ",[568,3644,3646,3663],{"className":3645},[571],[568,3647,3649],{"className":3648},[575],[577,3650,3651],{"xmlns":579},[581,3652,3653,3661],{},[584,3654,3655],{},[973,3656,3657,3659],{},[987,3658,3155],{},[595,3660,1068],{},[647,3662,3546],{"encoding":649},[568,3664,3666],{"className":3665,"ariaHidden":589},[654],[568,3667,3669,3672,3675,3681,3684],{"className":3668},[658],[568,3670],{"className":3671,"style":3482},[662],[568,3673,3395],{"className":3674},[667],[568,3676,3678],{"className":3677},[667],[568,3679,1696],{"className":3680},[1784],[568,3682,2962],{"className":3683},[667],[568,3685,3687,3690],{"className":3686},[667],[568,3688,3139],{"className":3689},[667],[568,3691,3693],{"className":3692},[1012],[568,3694,3696],{"className":3695},[672],[568,3697,3699],{"className":3698},[676],[568,3700,3702],{"className":3701,"style":1081},[680],[568,3703,3704,3707],{"style":1024},[568,3705],{"className":3706,"style":1028},[687],[568,3708,3710],{"className":3709},[1032,1033,1034,1035],[568,3711,1068],{"className":3712},[729,1035]," ablesen.",[1181,3715,3716],{},"Im Nenner die Längen addieren statt multiplizieren.",{"title":3718,"searchDepth":3719,"depth":3719,"links":3720},"",2,[3721,3722,3723],{"id":561,"depth":3719,"text":562},{"id":1663,"depth":3719,"text":1664},{"id":3439,"depth":3719,"text":3440},[3725,3728,3730,3733,3735,3737,3740,3743,3745],{"lhs":3726,"rhs":3727},"(1*0 + 0*1 + 1*1) \u002F (sqrt(1^2 + 0^2 + 1^2) * sqrt(0^2 + 1^2 + 1^2))","1\u002F2",{"lhs":3729,"rhs":2022},"acos(1\u002F2) * 180\u002Fpi",{"lhs":3731,"rhs":3732},"2*1 + 1*(-2) + (-2)*2","-4",{"lhs":3734,"rhs":2962},"sqrt(2^2 + 1^2 + (-2)^2)",{"lhs":3736,"rhs":2962},"sqrt(1^2 + (-2)^2 + 2^2)",{"lhs":3738,"rhs":3739},"round(acos(-4\u002F9) * 180\u002Fpi, 2)","116.39",{"lhs":3741,"rhs":3742},"round(acos(4\u002F9) * 180\u002Fpi, 2)","63.61",{"lhs":3744,"rhs":1128},"round(acos(-4\u002F9) * 180\u002Fpi + acos(4\u002F9) * 180\u002Fpi, 6)",{"lhs":3746,"rhs":3747},"round(acos(-4\u002F9), 2)","2.03","\\vec{a} \\neq \\vec{0},\\ \\vec{b} \\neq \\vec{0},\\ 0^\\circ \\leq \\varphi \\leq 180^\\circ","\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 mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmn>0\u003C\u002Fmn>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\vec{a} \\neq \\vec{0},\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:0.9084em;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 mathnormal\">a\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\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.1219em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9274em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2134em;\">\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=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmover accent=\"true\">\u003Cmi>b\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo mathvariant=\"normal\">≠\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmn>0\u003C\u002Fmn>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\vec{b} \\neq \\vec{0},\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:0.9774em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" 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=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">\u003Cspan class=\"mrel\">\u003Cspan class=\"mord katex-vbox\">\u003Cspan class=\"katex-thinbox\">\u003Cspan class=\"rlap\">\u003Cspan class=\"katex-strut\" style=\"height:0.8889em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-inner\">\u003Cspan class=\"mord\">\u003Cspan class=\"mrel\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-fix\">\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mrel\">=\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.2778em;\">\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan class=\"katex-base\">\u003Cspan class=\"katex-strut\" style=\"height:1.1219em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9274em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003C\u002Fspan>\u003Cspan style=\"top:-3.2134em;\">\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=\"mpunct\">,\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan> \u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmsup>\u003Cmn>0\u003C\u002Fmn>\u003Cmo>∘\u003C\u002Fmo>\u003C\u002Fmsup>\u003Cmo>≤\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo>≤\u003C\u002Fmo>\u003Cmsup>\u003Cmn>180\u003C\u002Fmn>\u003Cmo>∘\u003C\u002Fmo>\u003C\u002Fmsup>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">0^\\circ \\leq \\varphi \\leq 180^\\circ\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:0.8101em;vertical-align:-0.136em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6741em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">∘\u003C\u002Fspan>\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:0.8304em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\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:0.6741em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">18\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">0\u003C\u002Fspan>\u003Cspan class=\"msupsub\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.6741em;\">\u003Cspan style=\"top:-3.063em;margin-right:0.05em;\">\u003Cspan class=\"pstrut\" style=\"height:2.7em;\">\u003C\u002Fspan>\u003Cspan class=\"katex-sizing reset-size6 size3 mtight\">\u003Cspan class=\"mbin mtight\">∘\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>\u003C\u002Fspan>",null,"md","angle-between-vectors","\\cos\\varphi = \\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}| \\cdot |\\vec{b}|}","\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>\u003Cmi>cos\u003C\u002Fmi>\u003Cmo>⁡\u003C\u002Fmo>\u003Cmi>φ\u003C\u002Fmi>\u003Cmo>=\u003C\u002Fmo>\u003Cmfrac>\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>\u003C\u002Fmrow>\u003Cmrow>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmover accent=\"true\">\u003Cmi>a\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003Cmover accent=\"true\">\u003Cmi>b\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmi mathvariant=\"normal\">∣\u003C\u002Fmi>\u003C\u002Fmrow>\u003C\u002Fmfrac>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">\\cos\\varphi = \\frac{\\vec{a} \\cdot \\vec{b}}{|\\vec{a}| \\cdot |\\vec{b}|}\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:0.625em;vertical-align:-0.1944em;\">\u003C\u002Fspan>\u003Cspan class=\"mop\">cos\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1667em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\">φ\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:2.7719em;vertical-align:-1.1174em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mopen nulldelimiter\">\u003C\u002Fspan>\u003Cspan class=\"mfrac\">\u003Cspan class=\"vlist-t vlist-t2\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:1.6544em;\">\u003Cspan style=\"top:-2.1326em;\">\u003Cspan class=\"pstrut\" style=\"height:3em;\">\u003C\u002Fspan>\u003Cspan class=\"mord\">\u003Cspan class=\"mord\">∣\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\">a\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=\"mord\">∣\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\">∣\u003C\u002Fspan>\u003Cspan class=\"mord katex-accent\">\u003Cspan class=\"vlist-t\">\u003Cspan class=\"vlist-r\">\u003Cspan class=\"vlist\" style=\"height:0.9774em;\">\u003Cspan style=\"top:-3em;\">\u003Cspan class=\"pstrut\" 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\" 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