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in die Koordinatenform von ",[569,1212,1214,1227],{"className":1213},[572],[569,1215,1217],{"className":1216},[576],[578,1218,1219],{"xmlns":580},[582,1220,1221,1225],{},[585,1222,1223],{},[588,1224,590],{},[592,1226,590],{"encoding":594},[569,1228,1230],{"className":1229,"ariaHidden":599},[598],[569,1231,1233,1236],{"className":1232},[603],[569,1234],{"className":1235,"style":608},[607],[569,1237,590],{"className":1238,"style":614},[612,613]," einsetzen und ",[569,1241,1243,1256],{"className":1242},[572],[569,1244,1246],{"className":1245},[576],[578,1247,1248],{"xmlns":580},[582,1249,1250,1254],{},[585,1251,1252],{},[588,1253,959],{},[592,1255,959],{"encoding":594},[569,1257,1259],{"className":1258,"ariaHidden":599},[598],[569,1260,1262,1265],{"className":1261},[603],[569,1263],{"className":1264,"style":1153},[607],[569,1266,959],{"className":1267},[612,613]," berechnen.",[899,1270,1271,1272,1503,1504,891],{},"Der Lotfußpunkt ist ",[569,1273,1275,1314],{"className":1274},[572],[569,1276,1278],{"className":1277},[576],[578,1279,1280],{"xmlns":580},[582,1281,1282,1311],{},[585,1283,1284,1291,1293,1299,1301,1303,1305],{},[937,1285,1286,1289],{"accent":599},[588,1287,1288],{},"f",[635,1290,944],{},[635,1292,947],{},[937,1294,1295,1297],{"accent":599},[588,1296,565],{},[635,1298,944],{},[635,1300,956],{},[588,1302,959],{},[635,1304,962],{},[937,1306,1307,1309],{"accent":599},[588,1308,967],{},[635,1310,944],{},[592,1312,1313],{"encoding":594},"\\vec{f} = \\vec{p} + t \\cdot 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der Spiegelpunkt ",[569,1505,1507,1558],{"className":1506},[572],[569,1508,1510],{"className":1509},[576],[578,1511,1512],{"xmlns":580},[582,1513,1514,1555],{},[585,1515,1516,1522,1525,1531,1533,1539,1541,1545,1547,1549],{},[937,1517,1518,1520],{"accent":599},[588,1519,565],{},[635,1521,944],{},[927,1523,1524],{}," ",[629,1526,1527,1529],{},[585,1528],{},[635,1530,639],{"mathvariant":637,"lspace":638,"rspace":638},[635,1532,947],{},[937,1534,1535,1537],{"accent":599},[588,1536,565],{},[635,1538,944],{},[635,1540,956],{},[1542,1543,1544],"mn",{},"2",[588,1546,959],{},[635,1548,962],{},[937,1550,1551,1553],{"accent":599},[588,1552,967],{},[635,1554,944],{},[592,1556,1557],{"encoding":594},"\\vec{p}\\,' = \\vec{p} + 2t \\cdot \\vec{n}",[569,1559,1561,1657,1718,1740],{"className":1560,"ariaHidden":599},[598],[569,1562,1564,1568,1614,1617,1648,1651,1654],{"className":1563},[603],[569,1565],{"className":1566,"style":1567},[607],"height:0.9463em;vertical-align:-0.1944em;",[569,1569,1571],{"className":1570},[612,1020],[569,1572,1574,1606],{"className":1573},[667,1091],[569,1575,1577,1603],{"className":1576},[671],[569,1578,1580,1588],{"className":1579,"style":982},[675],[569,1581,1582,1585],{"style":1032},[569,1583],{"className":1584,"style":1036},[682],[569,1586,565],{"className":1587},[612,613],[569,1589,1590,1593],{"style":1032},[569,1591],{"className":1592,"style":1036},[682],[569,1594,1596],{"className":1595,"style":1114},[1048],[569,1597,1599],{"className":1598,"style":1054},[1053],[1056,1600,1601],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,1602],{"d":1067},[569,1604,1126],{"className":1605},[1125],[569,1607,1609],{"className":1608},[671],[569,1610,1612],{"className":1611,"style":1133},[675],[569,1613],{},[569,1615],{"className":1616,"style":1005},[918],[569,1618,1620,1622],{"className":1619},[612],[569,1621],{},[569,1623,1625],{"className":1624},[663],[569,1626,1628],{"className":1627},[667],[569,1629,1631],{"className":1630},[671],[569,1632,1634],{"className":1633,"style":652},[675],[569,1635,1636,1639],{"style":678},[569,1637],{"className":1638,"style":683},[682],[569,1640,1642],{"className":1641},[687,688,689,690],[569,1643,1645],{"className":1644},[612,690],[569,1646,639],{"className":1647},[612,690],[569,1649],{"className":1650,"style":1071},[918],[569,1652,947],{"className":1653},[1012],[569,1655],{"className":1656,"style":1071},[918],[569,1658,1660,1663,1709,1712,1715],{"className":1659},[603],[569,1661],{"className":1662,"style":1084},[607],[569,1664,1666],{"className":1665},[612,1020],[569,1667,1669,1701],{"className":1668},[667,1091],[569,1670,1672,1698],{"className":1671},[671],[569,1673,1675,1683],{"className":1674,"style":982},[675],[569,1676,1677,1680],{"style":1032},[569,1678],{"className":1679,"style":1036},[682],[569,1681,565],{"className":1682},[612,613],[569,1684,1685,1688],{"style":1032},[569,1686],{"className":1687,"style":1036},[682],[569,1689,1691],{"className":1690,"style":1114},[1048],[569,1692,1694],{"className":1693,"style":1054},[1053],[1056,1695,1696],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,1697],{"d":1067},[569,1699,1126],{"className":1700},[1125],[569,1702,1704],{"className":1703},[671],[569,1705,1707],{"className":1706,"style":1133},[675],[569,1708],{},[569,1710],{"className":1711,"style":1139},[918],[569,1713,956],{"className":1714},[1143],[569,1716],{"className":1717,"style":1139},[918],[569,1719,1721,1725,1728,1731,1734,1737],{"className":1720},[603],[569,1722],{"className":1723,"style":1724},[607],"height:0.6444em;",[569,1726,1544],{"className":1727},[612],[569,1729,959],{"className":1730},[612,613],[569,1732],{"className":1733,"style":1139},[918],[569,1735,962],{"className":1736},[1143],[569,1738],{"className":1739,"style":1139},[918],[569,1741,1743,1746],{"className":1742},[603],[569,1744],{"className":1745,"style":982},[607],[569,1747,1749],{"className":1748},[612,1020],[569,1750,1752],{"className":1751},[667],[569,1753,1755],{"className":1754},[671],[569,1756,1758,1766],{"className":1757,"style":982},[675],[569,1759,1760,1763],{"style":1032},[569,1761],{"className":1762,"style":1036},[682],[569,1764,967],{"className":1765},[612,613],[569,1767,1768,1771],{"style":1032},[569,1769],{"className":1770,"style":1036},[682],[569,1772,1774],{"className":1773,"style":1200},[1048],[569,1775,1777],{"className":1776,"style":1054},[1053],[1056,1778,1779],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,1780],{"d":1067},[565,1782,1783,1784,891],{},"Die Formel oben fasst diese Schritte zusammen, denn das Einsetzen ergibt ",[569,1785,1787,1846],{"className":1786},[572],[569,1788,1790],{"className":1789},[576],[578,1791,1792],{"xmlns":580},[582,1793,1794,1843],{},[585,1795,1796,1798,1800,1803],{},[588,1797,959],{},[635,1799,947],{},[635,1801,1802],{},"−",[1804,1805,1806,1827],"mfrac",{},[585,1807,1808,1814,1816,1822,1824],{},[937,1809,1810,1812],{"accent":599},[588,1811,967],{},[635,1813,944],{},[635,1815,962],{},[937,1817,1818,1820],{"accent":599},[588,1819,565],{},[635,1821,944],{},[635,1823,1802],{},[588,1825,1826],{},"d",[585,1828,1829,1835,1837],{},[937,1830,1831,1833],{"accent":599},[588,1832,967],{},[635,1834,944],{},[635,1836,962],{},[937,1838,1839,1841],{"accent":599},[588,1840,967],{},[635,1842,944],{},[592,1844,1845],{"encoding":594},"t = -\\frac{\\vec{n} \\cdot \\vec{p} - d}{\\vec{n} \\cdot \\vec{n}}",[569,1847,1849,1867],{"className":1848,"ariaHidden":599},[598],[569,1850,1852,1855,1858,1861,1864],{"className":1851},[603],[569,1853],{"className":1854,"style":1153},[607],[569,1856,959],{"className":1857},[612,613],[569,1859],{"className":1860,"style":1071},[918],[569,1862,947],{"className":1863},[1012],[569,1865],{"className":1866,"style":1071},[918],[569,1868,1870,1874,1877],{"className":1869},[603],[569,1871],{"className":1872,"style":1873},[607],"height:1.2909em;vertical-align:-0.345em;",[569,1875,1802],{"className":1876},[612],[569,1878,1880,1885,2110],{"className":1879},[612],[569,1881],{"className":1882},[1883,1884],"mopen","nulldelimiter",[569,1886,1888],{"className":1887},[1804],[569,1889,1891,2101],{"className":1890},[667,1091],[569,1892,1894,2098],{"className":1893},[671],[569,1895,1898,1985,1996],{"className":1896,"style":1897},[675],"height:0.9459em;",[569,1899,1901,1904],{"style":1900},"top:-2.655em;",[569,1902],{"className":1903,"style":1036},[682],[569,1905,1907],{"className":1906},[687,688,689,690],[569,1908,1910,1947,1950],{"className":1909},[612,690],[569,1911,1913],{"className":1912},[612,1020,690],[569,1914,1916],{"className":1915},[667],[569,1917,1919],{"className":1918},[671],[569,1920,1922,1932],{"className":1921,"style":982},[675],[569,1923,1925,1929],{"style":1924},"top:-2.714em;",[569,1926],{"className":1927,"style":1928},[682],"height:2.714em;",[569,1930,967],{"className":1931},[612,613,690],[569,1933,1934,1937],{"style":1924},[569,1935],{"className":1936,"style":1928},[682],[569,1938,1940],{"className":1939,"style":1200},[1048],[569,1941,1943],{"className":1942,"style":1054},[1053,690],[1056,1944,1945],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,1946],{"d":1067},[569,1948,962],{"className":1949},[1143,690],[569,1951,1953],{"className":1952},[612,1020,690],[569,1954,1956],{"className":1955},[667],[569,1957,1959],{"className":1958},[671],[569,1960,1962,1970],{"className":1961,"style":982},[675],[569,1963,1964,1967],{"style":1924},[569,1965],{"className":1966,"style":1928},[682],[569,1968,967],{"className":1969},[612,613,690],[569,1971,1972,1975],{"style":1924},[569,1973],{"className":1974,"style":1928},[682],[569,1976,1978],{"className":1977,"style":1200},[1048],[569,1979,1981],{"className":1980,"style":1054},[1053,690],[1056,1982,1983],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,1984],{"d":1067},[569,1986,1988,1991],{"style":1987},"top:-3.23em;",[569,1989],{"className":1990,"style":1036},[682],[569,1992],{"className":1993,"style":1995},[1994],"frac-line","border-bottom-width:0.04em;",[569,1997,1999,2002],{"style":1998},"top:-3.4461em;",[569,2000],{"className":2001,"style":1036},[682],[569,2003,2005],{"className":2004},[687,688,689,690],[569,2006,2008,2043,2046,2092,2095],{"className":2007},[612,690],[569,2009,2011],{"className":2010},[612,1020,690],[569,2012,2014],{"className":2013},[667],[569,2015,2017],{"className":2016},[671],[569,2018,2020,2028],{"className":2019,"style":982},[675],[569,2021,2022,2025],{"style":1924},[569,2023],{"className":2024,"style":1928},[682],[569,2026,967],{"className":2027},[612,613,690],[569,2029,2030,2033],{"style":1924},[569,2031],{"className":2032,"style":1928},[682],[569,2034,2036],{"className":2035,"style":1200},[1048],[569,2037,2039],{"className":2038,"style":1054},[1053,690],[1056,2040,2041],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,2042],{"d":1067},[569,2044,962],{"className":2045},[1143,690],[569,2047,2049],{"className":2048},[612,1020,690],[569,2050,2052,2084],{"className":2051},[667,1091],[569,2053,2055,2081],{"className":2054},[671],[569,2056,2058,2066],{"className":2057,"style":982},[675],[569,2059,2060,2063],{"style":1924},[569,2061],{"className":2062,"style":1928},[682],[569,2064,565],{"className":2065},[612,613,690],[569,2067,2068,2071],{"style":1924},[569,2069],{"className":2070,"style":1928},[682],[569,2072,2074],{"className":2073,"style":1114},[1048],[569,2075,2077],{"className":2076,"style":1054},[1053,690],[1056,2078,2079],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,2080],{"d":1067},[569,2082,1126],{"className":2083},[1125],[569,2085,2087],{"className":2086},[671],[569,2088,2090],{"className":2089,"style":1133},[675],[569,2091],{},[569,2093,1802],{"className":2094},[1143,690],[569,2096,1826],{"className":2097},[612,613,690],[569,2099,1126],{"className":2100},[1125],[569,2102,2104],{"className":2103},[671],[569,2105,2108],{"className":2106,"style":2107},[675],"height:0.345em;",[569,2109],{},[569,2111],{"className":2112},[2113,1884],"mclose",[565,2115,2116,2117,2145,2146,2303,2304,2388],{},"Liegt ",[569,2118,2120,2133],{"className":2119},[572],[569,2121,2123],{"className":2122},[576],[578,2124,2125],{"xmlns":580},[582,2126,2127,2131],{},[585,2128,2129],{},[588,2130,633],{},[592,2132,633],{"encoding":594},[569,2134,2136],{"className":2135,"ariaHidden":599},[598],[569,2137,2139,2142],{"className":2138},[603],[569,2140],{"className":2141,"style":608},[607],[569,2143,633],{"className":2144,"style":659},[612,613]," in der Ebene, ist ",[569,2147,2149,2179],{"className":2148},[572],[569,2150,2152],{"className":2151},[576],[578,2153,2154],{"xmlns":580},[582,2155,2156,2176],{},[585,2157,2158,2164,2166,2172,2174],{},[937,2159,2160,2162],{"accent":599},[588,2161,967],{},[635,2163,944],{},[635,2165,962],{},[937,2167,2168,2170],{"accent":599},[588,2169,565],{},[635,2171,944],{},[635,2173,947],{},[588,2175,1826],{},[592,2177,2178],{"encoding":594},"\\vec{n} \\cdot \\vec{p} = d",[569,2180,2182,2232,2293],{"className":2181,"ariaHidden":599},[598],[569,2183,2185,2188,2223,2226,2229],{"className":2184},[603],[569,2186],{"className":2187,"style":982},[607],[569,2189,2191],{"className":2190},[612,1020],[569,2192,2194],{"className":2193},[667],[569,2195,2197],{"className":2196},[671],[569,2198,2200,2208],{"className":2199,"style":982},[675],[569,2201,2202,2205],{"style":1032},[569,2203],{"className":2204,"style":1036},[682],[569,2206,967],{"className":2207},[612,613],[569,2209,2210,2213],{"style":1032},[569,2211],{"className":2212,"style":1036},[682],[569,2214,2216],{"className":2215,"style":1200},[1048],[569,2217,2219],{"className":2218,"style":1054},[1053],[1056,2220,2221],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,2222],{"d":1067},[569,2224],{"className":2225,"style":1139},[918],[569,2227,962],{"className":2228},[1143],[569,2230],{"className":2231,"style":1139},[918],[569,2233,2235,2238,2284,2287,2290],{"className":2234},[603],[569,2236],{"className":2237,"style":1084},[607],[569,2239,2241],{"className":2240},[612,1020],[569,2242,2244,2276],{"className":2243},[667,1091],[569,2245,2247,2273],{"className":2246},[671],[569,2248,2250,2258],{"className":2249,"style":982},[675],[569,2251,2252,2255],{"style":1032},[569,2253],{"className":2254,"style":1036},[682],[569,2256,565],{"className":2257},[612,613],[569,2259,2260,2263],{"style":1032},[569,2261],{"className":2262,"style":1036},[682],[569,2264,2266],{"className":2265,"style":1114},[1048],[569,2267,2269],{"className":2268,"style":1054},[1053],[1056,2270,2271],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,2272],{"d":1067},[569,2274,1126],{"className":2275},[1125],[569,2277,2279],{"className":2278},[671],[569,2280,2282],{"className":2281,"style":1133},[675],[569,2283],{},[569,2285],{"className":2286,"style":1071},[918],[569,2288,947],{"className":2289},[1012],[569,2291],{"className":2292,"style":1071},[918],[569,2294,2296,2300],{"className":2295},[603],[569,2297],{"className":2298,"style":2299},[607],"height:0.6944em;",[569,2301,1826],{"className":2302},[612,613]," und ",[569,2305,2307,2329],{"className":2306},[572],[569,2308,2310],{"className":2309},[576],[578,2311,2312],{"xmlns":580},[582,2313,2314,2326],{},[585,2315,2316,2322,2324],{},[629,2317,2318,2320],{},[588,2319,633],{},[635,2321,639],{"mathvariant":637,"lspace":638,"rspace":638},[635,2323,947],{},[588,2325,633],{},[592,2327,2328],{"encoding":594},"P' = P",[569,2330,2332,2379],{"className":2331,"ariaHidden":599},[598],[569,2333,2335,2338,2370,2373,2376],{"className":2334},[603],[569,2336],{"className":2337,"style":652},[607],[569,2339,2341,2344],{"className":2340},[612],[569,2342,633],{"className":2343,"style":659},[612,613],[569,2345,2347],{"className":2346},[663],[569,2348,2350],{"className":2349},[667],[569,2351,2353],{"className":2352},[671],[569,2354,2356],{"className":2355,"style":652},[675],[569,2357,2358,2361],{"style":678},[569,2359],{"className":2360,"style":683},[682],[569,2362,2364],{"className":2363},[687,688,689,690],[569,2365,2367],{"className":2366},[612,690],[569,2368,639],{"className":2369},[612,690],[569,2371],{"className":2372,"style":1071},[918],[569,2374,947],{"className":2375},[1012],[569,2377],{"className":2378,"style":1071},[918],[569,2380,2382,2385],{"className":2381},[603],[569,2383],{"className":2384,"style":608},[607],[569,2386,633],{"className":2387,"style":659},[612,613],". Um eine Gerade oder ein Dreieck zu spiegeln, spiegelt man zwei bzw. drei Punkte.",[560,2390,2392],{"id":2391},"beispiele","Beispiele",[565,2394,2395,2303,2673,2767,2768,3034],{},[569,2396,2398,2458],{"className":2397},[572],[569,2399,2401],{"className":2400},[576],[578,2402,2403],{"xmlns":580},[582,2404,2405,2455],{},[585,2406,2407,2409,2411,2413,2415,2417,2419,2421,2429,2431,2433,2439,2441,2443,2450,2452],{},[588,2408,590],{},[918,2410],{},[918,2412],{"width":922},[635,2414],{"lspace":638,"rspace":925},[927,2416,929],{},[635,2418,932],{"lspace":638,"rspace":638},[918,2420],{"width":935},[2422,2423,2424,2426],"msub",{},[588,2425,941],{},[1542,2427,2428],{},"1",[635,2430,956],{},[1542,2432,1544],{},[2422,2434,2435,2437],{},[588,2436,941],{},[1542,2438,1544],{},[635,2440,956],{},[1542,2442,1544],{},[2422,2444,2445,2447],{},[588,2446,941],{},[1542,2448,2449],{},"3",[635,2451,947],{},[1542,2453,2454],{},"9",[592,2456,2457],{"encoding":594},"E\\colon x_1 + 2x_2 + 2x_3 = 9",[569,2459,2461,2547,2606,2664],{"className":2460,"ariaHidden":599},[598],[569,2462,2464,2468,2471,2474,2477,2480,2483,2486,2492,2495,2538,2541,2544],{"className":2463},[603],[569,2465],{"className":2466,"style":2467},[607],"height:0.8333em;vertical-align:-0.15em;",[569,2469,590],{"className":2470,"style":614},[612,613],[569,2472],{"className":2473},[918,989],[569,2475],{"className":2476,"style":993},[918],[569,2478],{"className":2479},[997],[569,2481],{"className":2482,"style":1001},[918],[569,2484],{"className":2485,"style":1005},[918],[569,2487,2489],{"className":2488},[612],[569,2490,932],{"className":2491},[1012],[569,2493],{"className":2494,"style":1016},[918],[569,2496,2498,2501],{"className":2497},[612],[569,2499,941],{"className":2500},[612,613],[569,2502,2504],{"className":2503},[663],[569,2505,2507,2529],{"className":2506},[667,1091],[569,2508,2510,2526],{"className":2509},[671],[569,2511,2514],{"className":2512,"style":2513},[675],"height:0.3011em;",[569,2515,2517,2520],{"style":2516},"top:-2.55em;margin-left:0em;margin-right:0.05em;",[569,2518],{"className":2519,"style":683},[682],[569,2521,2523],{"className":2522},[687,688,689,690],[569,2524,2428],{"className":2525},[612,690],[569,2527,1126],{"className":2528},[1125],[569,2530,2532],{"className":2531},[671],[569,2533,2536],{"className":2534,"style":2535},[675],"height:0.15em;",[569,2537],{},[569,2539],{"className":2540,"style":1139},[918],[569,2542,956],{"className":2543},[1143],[569,2545],{"className":2546,"style":1139},[918],[569,2548,2550,2554,2557,2597,2600,2603],{"className":2549},[603],[569,2551],{"className":2552,"style":2553},[607],"height:0.7944em;vertical-align:-0.15em;",[569,2555,1544],{"className":2556},[612],[569,2558,2560,2563],{"className":2559},[612],[569,2561,941],{"className":2562},[612,613],[569,2564,2566],{"className":2565},[663],[569,2567,2569,2589],{"className":2568},[667,1091],[569,2570,2572,2586],{"className":2571},[671],[569,2573,2575],{"className":2574,"style":2513},[675],[569,2576,2577,2580],{"style":2516},[569,2578],{"className":2579,"style":683},[682],[569,2581,2583],{"className":2582},[687,688,689,690],[569,2584,1544],{"className":2585},[612,690],[569,2587,1126],{"className":2588},[1125],[569,2590,2592],{"className":2591},[671],[569,2593,2595],{"className":2594,"style":2535},[675],[569,2596],{},[569,2598],{"className":2599,"style":1139},[918],[569,2601,956],{"className":2602},[1143],[569,2604],{"className":2605,"style":1139},[918],[569,2607,2609,2612,2615,2655,2658,2661],{"className":2608},[603],[569,2610],{"className":2611,"style":2553},[607],[569,2613,1544],{"className":2614},[612],[569,2616,2618,2621],{"className":2617},[612],[569,2619,941],{"className":2620},[612,613],[569,2622,2624],{"className":2623},[663],[569,2625,2627,2647],{"className":2626},[667,1091],[569,2628,2630,2644],{"className":2629},[671],[569,2631,2633],{"className":2632,"style":2513},[675],[569,2634,2635,2638],{"style":2516},[569,2636],{"className":2637,"style":683},[682],[569,2639,2641],{"className":2640},[687,688,689,690],[569,2642,2449],{"className":2643},[612,690],[569,2645,1126],{"className":2646},[1125],[569,2648,2650],{"className":2649},[671],[569,2651,2653],{"className":2652,"style":2535},[675],[569,2654],{},[569,2656],{"className":2657,"style":1071},[918],[569,2659,947],{"className":2660},[1012],[569,2662],{"className":2663,"style":1071},[918],[569,2665,2667,2670],{"className":2666},[603],[569,2668],{"className":2669,"style":1724},[607],[569,2671,2454],{"className":2672},[612],[569,2674,2676,2709],{"className":2675},[572],[569,2677,2679],{"className":2678},[576],[578,2680,2681],{"xmlns":580},[582,2682,2683,2706],{},[585,2684,2685,2687,2691,2693,2696,2698,2700,2703],{},[588,2686,633],{},[635,2688,2690],{"stretchy":2689},"false","(",[1542,2692,1544],{},[635,2694,2695],{},"∣",[1542,2697,2449],{},[635,2699,2695],{},[1542,2701,2702],{},"5",[635,2704,2705],{"stretchy":2689},")",[592,2707,2708],{"encoding":594},"P(2 \\mid 3 \\mid 5)",[569,2710,2712,2737,2755],{"className":2711,"ariaHidden":599},[598],[569,2713,2715,2719,2722,2725,2728,2731,2734],{"className":2714},[603],[569,2716],{"className":2717,"style":2718},[607],"height:1em;vertical-align:-0.25em;",[569,2720,633],{"className":2721,"style":659},[612,613],[569,2723,2690],{"className":2724},[1883],[569,2726,1544],{"className":2727},[612],[569,2729],{"className":2730,"style":1071},[918],[569,2732,2695],{"className":2733},[1012],[569,2735],{"className":2736,"style":1071},[918],[569,2738,2740,2743,2746,2749,2752],{"className":2739},[603],[569,2741],{"className":2742,"style":2718},[607],[569,2744,2449],{"className":2745},[612],[569,2747],{"className":2748,"style":1071},[918],[569,2750,2695],{"className":2751},[1012],[569,2753],{"className":2754,"style":1071},[918],[569,2756,2758,2761,2764],{"className":2757},[603],[569,2759],{"className":2760,"style":2718},[607],[569,2762,2702],{"className":2763},[612],[569,2765,2705],{"className":2766},[2113],". Lotgerade ",[569,2769,2771,2849],{"className":2770},[572],[569,2772,2774],{"className":2773},[576],[578,2775,2776],{"xmlns":580},[582,2777,2778,2846],{},[585,2779,2780,2782,2784,2786,2788,2790,2792,2794,2800,2802,2804,2806,2809,2812,2814,2816,2818,2820,2822,2824,2826,2828,2830,2832,2834,2836,2838,2840,2842,2844],{},[588,2781,916],{"mathvariant":637},[918,2783],{},[918,2785],{"width":922},[635,2787],{"lspace":638,"rspace":925},[927,2789,929],{},[635,2791,932],{"lspace":638,"rspace":638},[918,2793],{"width":935},[937,2795,2796,2798],{"accent":599},[588,2797,941],{},[635,2799,944],{},[635,2801,947],{},[635,2803,2690],{"stretchy":2689},[1542,2805,1544],{},[635,2807,2808],{"separator":599},",",[927,2810,2811],{}," ",[1542,2813,2449],{},[635,2815,2808],{"separator":599},[927,2817,2811],{},[1542,2819,2702],{},[635,2821,2705],{"stretchy":2689},[635,2823,956],{},[588,2825,959],{},[635,2827,962],{},[635,2829,2690],{"stretchy":2689},[1542,2831,2428],{},[635,2833,2808],{"separator":599},[927,2835,2811],{},[1542,2837,1544],{},[635,2839,2808],{"separator":599},[927,2841,2811],{},[1542,2843,1544],{},[635,2845,2705],{"stretchy":2689},[592,2847,2848],{"encoding":594},"\\ell\\colon \\vec{x} = (2,\\ 3,\\ 5) + t \\cdot (1,\\ 2,\\ 2)",[569,2850,2852,2929,2977,2995],{"className":2851,"ariaHidden":599},[598],[569,2853,2855,2858,2861,2864,2867,2870,2873,2876,2882,2885,2920,2923,2926],{"className":2854},[603],[569,2856],{"className":2857,"style":982},[607],[569,2859,916],{"className":2860},[612],[569,2862],{"className":2863},[918,989],[569,2865],{"className":2866,"style":993},[918],[569,2868],{"className":2869},[997],[569,2871],{"className":2872,"style":1001},[918],[569,2874],{"className":2875,"style":1005},[918],[569,2877,2879],{"className":2878},[612],[569,2880,932],{"className":2881},[1012],[569,2883],{"className":2884,"style":1016},[918],[569,2886,2888],{"className":2887},[612,1020],[569,2889,2891],{"className":2890},[667],[569,2892,2894],{"className":2893},[671],[569,2895,2897,2905],{"className":2896,"style":982},[675],[569,2898,2899,2902],{"style":1032},[569,2900],{"className":2901,"style":1036},[682],[569,2903,941],{"className":2904},[612,613],[569,2906,2907,2910],{"style":1032},[569,2908],{"className":2909,"style":1036},[682],[569,2911,2913],{"className":2912,"style":1049},[1048],[569,2914,2916],{"className":2915,"style":1054},[1053],[1056,2917,2918],{"xmlns":1058,"width":1059,"height":1060,"style":1061,"viewBox":1062,"preserveAspectRatio":1063},[1065,2919],{"d":1067},[569,2921],{"className":2922,"style":1071},[918],[569,2924,947],{"className":2925},[1012],[569,2927],{"className":2928,"style":1071},[918],[569,2930,2932,2935,2938,2941,2944,2947,2950,2953,2956,2959,2962,2965,2968,2971,2974],{"className":2931},[603],[569,2933],{"className":2934,"style":2718},[607],[569,2936,2690],{"className":2937},[1883],[569,2939,1544],{"className":2940},[612],[569,2942,2808],{"className":2943},[997],[569,2945,2811],{"className":2946},[918],[569,2948],{"className":2949,"style":1005},[918],[569,2951,2449],{"className":2952},[612],[569,2954,2808],{"className":2955},[997],[569,2957,2811],{"className":2958},[918],[569,2960],{"className":2961,"style":1005},[918],[569,2963,2702],{"className":2964},[612],[569,2966,2705],{"className":2967},[2113],[569,2969],{"className":2970,"style":1139},[918],[569,2972,956],{"className":2973},[1143],[569,2975],{"className":2976,"style":1139},[918],[569,2978,2980,2983,2986,2989,2992],{"className":2979},[603],[569,2981],{"className":2982,"style":1153},[607],[569,2984,959],{"className":2985},[612,613],[569,2987],{"className":2988,"style":1139},[918],[569,2990,962],{"className":2991},[1143],[569,2993],{"className":2994,"style":1139},[918],[569,2996,2998,3001,3004,3007,3010,3013,3016,3019,3022,3025,3028,3031],{"className":2997},[603],[569,2999],{"className":3000,"style":2718},[607],[569,3002,2690],{"className":3003},[1883],[569,3005,2428],{"className":3006},[612],[569,3008,2808],{"className":3009},[997],[569,3011,2811],{"className":3012},[918],[569,3014],{"className":3015,"style":1005},[918],[569,3017,1544],{"className":3018},[612],[569,3020,2808],{"className":3021},[997],[569,3023,2811],{"className":3024},[918],[569,3026],{"className":3027,"style":1005},[918],[569,3029,1544],{"className":3030},[612],[569,3032,2705],{"className":3033},[2113],". Einsetzen:",[569,3036,3039],{"className":3037},[3038],"katex-display",[569,3040,3042,3128],{"className":3041},[572],[569,3043,3045],{"className":3044},[576],[578,3046,3048],{"xmlns":580,"display":3047},"block",[582,3049,3050,3125],{},[585,3051,3052,3054,3056,3058,3060,3062,3064,3066,3068,3070,3072,3074,3076,3078,3080,3082,3084,3086,3088,3090,3092,3094,3096,3099,3101,3103,3105,3107,3109,3112,3115,3117,3119,3121,3123],{},[635,3053,2690],{"stretchy":2689},[1542,3055,1544],{},[635,3057,956],{},[588,3059,959],{},[635,3061,2705],{"stretchy":2689},[635,3063,956],{},[1542,3065,1544],{},[635,3067,2690],{"stretchy":2689},[1542,3069,2449],{},[635,3071,956],{},[1542,3073,1544],{},[588,3075,959],{},[635,3077,2705],{"stretchy":2689},[635,3079,956],{},[1542,3081,1544],{},[635,3083,2690],{"stretchy":2689},[1542,3085,2702],{},[635,3087,956],{},[1542,3089,1544],{},[588,3091,959],{},[635,3093,2705],{"stretchy":2689},[635,3095,947],{},[1542,3097,3098],{},"18",[635,3100,956],{},[1542,3102,2454],{},[588,3104,959],{},[635,3106,947],{},[1542,3108,2454],{},[927,3110,3111],{},"  ",[635,3113,3114],{},"⇒",[927,3116,3111],{},[588,3118,959],{},[635,3120,947],{},[635,3122,1802],{},[1542,3124,2428],{},[592,3126,3127],{"encoding":594},"(2 + t) + 2(3 + 2t) + 2(5 + 2t) = 18 + 9t = 9 \\;\\Rightarrow\\; t = 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0v40h399900v-40z",[569,4710,2695],{"className":4711},[612],[569,4713],{"className":4714,"style":1071},[918],[569,4716,947],{"className":4717},[1012],[569,4719],{"className":4720,"style":1071},[918],[569,4722,4724,4728,4809,4812,4815],{"className":4723},[603],[569,4725],{"className":4726,"style":4727},[607],"height:1.04em;vertical-align:-0.1744em;",[569,4729,4732],{"className":4730},[612,4731],"sqrt",[569,4733,4735,4800],{"className":4734},[667,1091],[569,4736,4738,4797],{"className":4737},[671],[569,4739,4742,4779],{"className":4740,"style":4741},[675],"height:0.8656em;",[569,4743,4745,4748],{"className":4744,"style":1032},[4690],[569,4746],{"className":4747,"style":1036},[682],[569,4749,4752,4755,4758,4761,4764,4767,4770,4773,4776],{"className":4750,"style":4751},[612],"padding-left:0.833em;",[569,4753,4599],{"className":4754},[612],[569,4756],{"className":4757,"style":1139},[918],[569,4759,956],{"className":4760},[1143],[569,4762],{"className":4763,"style":1139},[918],[569,4765,4604],{"className":4766},[612],[569,4768],{"className":4769,"style":1139},[918],[569,4771,956],{"className":4772},[1143],[569,4774],{"className":4775,"style":1139},[918],[569,4777,4604],{"className":4778},[612],[569,4780,4782,4785],{"style":4781},"top:-2.8256em;",[569,4783],{"className":4784,"style":1036},[682],[569,4786,4789],{"className":4787,"style":4788},[4698],"min-width:0.853em;height:1.08em;",[1056,4790,4794],{"xmlns":1058,"width":4702,"height":4791,"viewBox":4792,"preserveAspectRatio":4793},"1.08em","0 0 400000 1080","xMinYMin slice",[1065,4795],{"d":4796},"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",[569,4798,1126],{"className":4799},[1125],[569,4801,4803],{"className":4802},[671],[569,4804,4807],{"className":4805,"style":4806},[675],"height:0.1744em;",[569,4808],{},[569,4810],{"className":4811,"style":1071},[918],[569,4813,947],{"className":4814},[1012],[569,4816],{"className":4817,"style":1071},[918],[569,4819,4821,4824],{"className":4820},[603],[569,4822],{"className":4823,"style":1724},[607],[569,4825,3477],{"className":4826},[612],[560,4828,4830],{"id":4829},"typische-fehler","Typische Fehler",[4832,4833,4834,4900,4931],"ul",{},[899,4835,4836,4837,4865,4866,891],{},"Nur bis zum Lotfußpunkt gehen. ",[569,4838,4840,4853],{"className":4839},[572],[569,4841,4843],{"className":4842},[576],[578,4844,4845],{"xmlns":580},[582,4846,4847,4851],{},[585,4848,4849],{},[588,4850,808],{},[592,4852,808],{"encoding":594},[569,4854,4856],{"className":4855,"ariaHidden":599},[598],[569,4857,4859,4862],{"className":4858},[603],[569,4860],{"className":4861,"style":608},[607],[569,4863,808],{"className":4864,"style":659},[612,613]," ist der Mittelpunkt, der Spiegelpunkt liegt bei ",[569,4867,4869,4885],{"className":4868},[572],[569,4870,4872],{"className":4871},[576],[578,4873,4874],{"xmlns":580},[582,4875,4876,4882],{},[585,4877,4878,4880],{},[1542,4879,1544],{},[588,4881,959],{},[592,4883,4884],{"encoding":594},"2t",[569,4886,4888],{"className":4887,"ariaHidden":599},[598],[569,4889,4891,4894,4897],{"className":4890},[603],[569,4892],{"className":4893,"style":1724},[607],[569,4895,1544],{"className":4896},[612],[569,4898,959],{"className":4899},[612,613],[899,4901,4902,4903,891],{},"Mit einem Richtungsvektor der Ebene statt mit dem Normalenvektor arbeiten. Die Lotgerade steht senkrecht auf ",[569,4904,4906,4919],{"className":4905},[572],[569,4907,4909],{"className":4908},[576],[578,4910,4911],{"xmlns":580},[582,4912,4913,4917],{},[585,4914,4915],{},[588,4916,590],{},[592,4918,590],{"encoding":594},[569,4920,4922],{"className":4921,"ariaHidden":599},[598],[569,4923,4925,4928],{"className":4924},[603],[569,4926],{"className":4927,"style":608},[607],[569,4929,590],{"className":4930,"style":614},[612,613],[899,4932,4933,4934,4962,4963,4991],{},"Beim Einsetzen die Ebenengleichung nicht vollständig nach ",[569,4935,4937,4950],{"className":4936},[572],[569,4938,4940],{"className":4939},[576],[578,4941,4942],{"xmlns":580},[582,4943,4944,4948],{},[585,4945,4946],{},[588,4947,959],{},[592,4949,959],{"encoding":594},[569,4951,4953],{"className":4952,"ariaHidden":599},[598],[569,4954,4956,4959],{"className":4955},[603],[569,4957],{"className":4958,"style":1153},[607],[569,4960,959],{"className":4961},[612,613]," auflösen, z.B. die Konstante ",[569,4964,4966,4979],{"className":4965},[572],[569,4967,4969],{"className":4968},[576],[578,4970,4971],{"xmlns":580},[582,4972,4973,4977],{},[585,4974,4975],{},[588,4976,1826],{},[592,4978,1826],{"encoding":594},[569,4980,4982],{"className":4981,"ariaHidden":599},[598],[569,4983,4985,4988],{"className":4984},[603],[569,4986],{"className":4987,"style":2299},[607],[569,4989,1826],{"className":4990},[612,613]," auf der falschen Seite stehen lassen.",{"title":4993,"searchDepth":4994,"depth":4994,"links":4995},"",2,[4996,4997,4998],{"id":562,"depth":4994,"text":563},{"id":2391,"depth":4994,"text":2392},{"id":4829,"depth":4994,"text":4830},[5000,5002,5004,5008,5010,5012,5015,5017,5019],{"lhs":5001,"rhs":1826},"n1*(p1 - (n1*p1 + n2*p2 + n3*p3 - d)\u002F(n1^2 + n2^2 + n3^2)*n1) + n2*(p2 - (n1*p1 + n2*p2 + n3*p3 - d)\u002F(n1^2 + n2^2 + n3^2)*n2) + n3*(p3 - (n1*p1 + n2*p2 + n3*p3 - d)\u002F(n1^2 + n2^2 + n3^2)*n3)",{"lhs":5003,"rhs":3098},"1*2 + 2*3 + 2*5",{"lhs":5005,"rhs":2454,"where":5006},"18 + 9*t",{"t":5007},-1,{"lhs":5009,"rhs":2454},"1 + 2*1 + 2*3",{"lhs":5011,"rhs":3663},"2 - 2*(18 - 9)\u002F9*1",{"lhs":5013,"rhs":5014},"3 - 2*(18 - 9)\u002F9*2","-1",{"lhs":5016,"rhs":2428},"5 - 2*(18 - 9)\u002F9*2",{"lhs":5018,"rhs":2449},"abs(18 - 9)\u002Fsqrt(1^2 + 2^2 + 2^2)",{"lhs":5020,"rhs":3477},"sqrt((0 - 2)^2 + (-1 - 3)^2 + (1 - 5)^2)","E\\colon \\vec{n} \\cdot \\vec{x} = d,\\ \\vec{n} \\neq \\vec{0}","\u003Cspan class=\"katex\">\u003Cspan class=\"katex-mathml\">\u003Cmath xmlns=\"http:\u002F\u002Fwww.w3.org\u002F1998\u002FMath\u002FMathML\">\u003Csemantics>\u003Cmrow>\u003Cmi>E\u003C\u002Fmi>\u003Cmspace>\u003C\u002Fmspace>\u003Cmspace width=\"0.1111em\"\u002F>\u003Cmo lspace=\"0em\" rspace=\"0.17em\">\u003C\u002Fmo>\u003Cmtext> ⁣\u003C\u002Fmtext>\u003Cmo lspace=\"0em\" rspace=\"0em\">:\u003C\u002Fmo>\u003Cmspace width=\"0.3333em\"\u002F>\u003Cmover accent=\"true\">\u003Cmi>n\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>⋅\u003C\u002Fmo>\u003Cmover accent=\"true\">\u003Cmi>x\u003C\u002Fmi>\u003Cmo>⃗\u003C\u002Fmo>\u003C\u002Fmover>\u003Cmo>=\u003C\u002Fmo>\u003Cmi>d\u003C\u002Fmi>\u003Cmo separator=\"true\">,\u003C\u002Fmo>\u003C\u002Fmrow>\u003Cannotation encoding=\"application\u002Fx-tex\">E\\colon \\vec{n} \\cdot \\vec{x} = d,\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:0.714em;\">\u003C\u002Fspan>\u003Cspan class=\"mord mathnormal\" style=\"margin-right:0.0576em;\">E\u003C\u002Fspan>\u003Cspan class=\"mspace nobreak\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:0.1111em;\">\u003C\u002Fspan>\u003Cspan class=\"mpunct\">\u003C\u002Fspan>\u003Cspan class=\"mspace\" style=\"margin-right:-0.1667em;\">\u003C\u002Fspan>\u003Cspan 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