{"id":1032,"date":"2024-01-30T12:17:53","date_gmt":"2024-01-30T06:17:53","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=1032"},"modified":"2024-11-26T17:54:11","modified_gmt":"2024-11-26T11:54:11","slug":"geometric-representation","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/geometric-representation\/","title":{"rendered":"\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09af\u09cb\u0997, \u09ac\u09bf\u09af\u09bc\u09cb\u0997 \u0993 \u0997\u09c1\u09a3\u09c7\u09b0 \u099c\u09cd\u09af\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa (Geometric representation of the addition, subtraction, multiplication of complex number)"},"content":{"rendered":"\r\n\r\n\r\n

\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u099c\u09cd\u09af\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa (Geometric representation of Complex Number)<\/span><\/h2>\r\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u0986\u09b0\u09cd\u0997\u09a8\u09cd\u09a1 \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 P<\/span><\/span> \u0993 Q<\/span><\/span> \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 z_{1}=x_{1}+i y_{1}<\/span><\/span> \u098f\u09ac\u0982 z_{2}=x_{2}+i y_{2}<\/span><\/span>, \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be<\/a><\/span> \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964 \u09a4\u09be\u09b9\u09b2\u09c7, P<\/span><\/span> \u0993 Q<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 \\left(x_{1}, y_{1}\\right)<\/span><\/span> \u098f\u09ac\u0982 \\left(x_{2}, y_{2}\\right)<\/span><\/span>.<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

OP<\/span> \u098f\u09ac\u0982 OQ<\/span> \u0995\u09c7 \u09b8\u09a8\u09cd\u09a8\u09bf\u09b9\u09bf\u09a4 \u09ac\u09be\u09b9\u09c1 \u09a7\u09b0\u09c7 OPRQ<\/span> \u09b8\u09be\u09ae\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf\u0964 O, R<\/span> \u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964 P, Q<\/span> \u0993 R<\/span> \u09b9\u09a4\u09c7 OX<\/span> \u098f\u09b0 \u0989\u09aa\u09b0 \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 PL, QM<\/span> \u0993 RN<\/span> \u09b2\u09ae\u09cd\u09ac \u0986\u0981\u0995\u09bf\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\r\n
\"Complex<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 OR<\/span><\/span> \u098f\u09b0 \u0985\u09ad\u09bf\u0995\u09cd\u09b7\u09c7\u09aa \u0990 \u098f\u0995\u0987 \u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 OP<\/span><\/span> \u098f\u09ac\u0982 PR<\/span><\/span>\u00a0\u0985\u09a5\u09ac\u09be <\/span>OQ\u00a0[\\because OQ||PR<\/span><\/span>\u098f\u09ac\u0982 OQ=PR]<\/span><\/span>-\u098f\u09b0 \u0985\u09ad\u09bf\u0995\u09cd\u09b7\u09c7\u09aa\u09c7\u09b0 \u09ac\u09c0\u099c\u0997\u09a3\u09bf\u09a4\u09c0\u09af\u09bc \u09af\u09cb\u0997\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964 \u09b8\u09c1\u09a4\u09b0\u09be\u0982 R<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 \\left(x_{1}+x_{2}, y_{1}+y_{2}\\right)<\/span><\/span>\u0964 \u0985\u09a4\u098f\u09ac, R<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \\left(x_{1}+x_{2}\\right)+i\\left(y_{1}+y_{2}\\right)<\/span><\/span>\u00a0\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0995\u09bf\u09a8\u09cd\u09a4\u09c1 z_{1}+z_{2}=x_{1}+i y_{1}+x_{2}+i y_{2}=\\left(x_{1}+x_{2}\\right)+i\\left(y_{1}+y_{2}\\right)<\/span><\/span>\u0964 \u0985\u09a4\u098f\u09ac, R<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 P<\/span><\/span> \u0993 Q<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6\u09bf\u09a4 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be<\/a><\/span> \u09a6\u09c1\u0987\u099f\u09bf\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8 P<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09bf\u09af\u09bc\u09c7 Q O || P S<\/span><\/span> \u0986\u0981\u0995\u09bf \u09af\u09c7\u09a8 Q O = P S<\/span><\/span>\u00a0<\/span>\u09b9\u09af\u09bc\u0964 <\/span>O, S<\/span><\/span> \u09af\u09cb\u0997 \u0995\u09b0\u09bf\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 S<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 z_{1}-z_{2}<\/span><\/span>, \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09ac\u09c7 \u098f\u09ac\u0982 \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09a6\u09c1\u0987\u099f\u09bf \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09b8\u09c2\u099a\u09bf\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09a6\u09c2\u09b0\u09a4\u09cd\u09ac =Q P=O S=\\left|z_{1}-z_{2}\\right|=\\left|\\left(x_{1}-x_{2}\\right)+i\\left(y_{1}-y_{2}\\right)\\right|=\\sqrt{\\left(x_{1}-x_{2}\\right)^{2}+\\left(y_{1}-y_{2}\\right)^{2}}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0997\u09c1\u09a3\u09c7\u09b0 \u099c\u09cd\u09af\u09be\u09ae\u09bf\u09a4\u09bf\u0995 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa (<\/b>Geometrical representation of multiplication of Complex Number<\/b>)<\/b><\/span><\/h2>\r\n\r\n\r\n\r\n
\r\n
\"Complex<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, P<\/span> \u0993 Q<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 z_1<\/span> \u0993 z_2<\/span> \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09a6\u09cd\u09ac\u09af\u09bc \u098f\u09ac\u0982 R<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09cd\u09ac\u09be\u09b0\u09be z=z_1z_2<\/span> \u09b8\u0982\u0996\u09cd\u09af\u09be\u099f\u09bf \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09c7\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u09ae\u09a1\u09c1\u09b2\u09be\u09b8 |z|=OR=OP\\times OQ<\/span> \u098f\u09ac\u0982 z<\/span> \u098f\u09b0 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u0997\u09c1\u09a3\u09ab\u09b2\u09c7\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f \\angle R O X=\\angle P O X+\\angle Q O X<\/span>\u0964 \u09ae\u09a8\u09c7 \u0995\u09b0\u09bf OX<\/span> \u09ac\u09b0\u09be\u09ac\u09b0 OA<\/span>\u00a0\u09b0\u09c7\u0996\u09be\u0982\u09b6 \u098f\u0995\u0995 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09c7\u0964 \u09a4\u09be\u09b9\u09b2\u09c7,<\/span><\/p>\r\n\r\n\r\n\r\n

\\frac{OR}{OQ}=\\frac{OP}{OA}<\/span> \u098f\u09ac\u0982 \\angle ROQ=\\angle ROX-\\angle QOX=\\angle POX=\\angle POA<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 POA<\/span><\/span> \u098f\u09ac\u0982 ROQ<\/span><\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09a6\u09cd\u09ac\u09af\u09bc \u09b8\u09a6\u09c3\u09b6\u0964 \u09a4\u09be\u0987 \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u0997\u09c1\u09a3\u09a8\u09c7\u09b0 \u09aa\u09cd\u09b0\u09a4\u09bf\u09b0\u09c2\u09aa \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09a8\u09bf\u09b0\u09cd\u09a3\u09af\u09bc \u0995\u09b0\u09a4\u09c7 \u09b9\u09b2\u09c7 OX<\/span><\/span> \u09ac\u09b0\u09be\u09ac\u09b0 \u098f\u0995\u0995\u00a0<\/span>\u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af\u09ac\u09bf\u09b6\u09bf\u09b7\u09cd\u099f OA<\/span><\/span> \u0985\u0982\u09b6 \u0995\u09c7\u099f\u09c7 \u09a8\u09bf\u09af\u09bc\u09c7 OQ<\/span><\/span> \u098f\u09b0 \u0989\u09aa\u09b0 OQ<\/span><\/span> \u098f\u09b0 \u09af\u09c7 \u09aa\u09be\u09b6\u09c7 OP<\/span><\/span> \u09a4\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b6\u09c7 \u098f\u09b0\u09c2\u09aa \u098f\u0995\u099f\u09bf OQR<\/span><\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09a4\u09c7 \u09b9\u09ac\u09c7 \u09af\u09be OPA<\/span><\/span> \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09c7\u09b0 \u09b8\u09a6\u09c3\u09b6\u0964 \u09a4\u09be\u09b9\u09b2\u09c7 R<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a6\u09c1\u0987\u099f\u09bf z_1<\/span><\/span> \u0993 z_2<\/span><\/span> \u098f\u09b0 \u0997\u09c1\u09a3\u09ab\u09b2 z<\/span><\/span> \u09b8\u09c2\u099a\u09bf\u09a4 \u0995\u09b0\u09ac\u09c7\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09b0\u09cd\u09a5\u09be\u09ce, \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be\u09b0 \u09ad\u09be\u0997\u09ab\u09b2\u09c7\u09b0 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f \u09a4\u09be\u09a6\u09c7\u09b0 \u09aa\u09c3\u09a5\u0995 \u09aa\u09c3\u09a5\u0995 \u09ad\u09be\u09ac\u09c7 \u0986\u09b0\u09cd\u0997\u09c1\u09ae\u09c7\u09a8\u09cd\u099f\u09c7\u09b0 \u09ac\u09bf\u09af\u09bc\u09cb\u0997\u09ab\u09b2\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u099c\u09cd\u09af\u09be\u09ae\u09bf\u09a4\u09bf\u0995<\/b> \u09ac\u09cd\u09af\u09be\u0996\u09cd\u09af\u09be<\/b><\/span>:<\/span> <\/span><\/h3>\r\n\r\n\r\n\r\n

\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u099a\u09bf\u09a4\u09cd\u09b0\u09c7 P<\/span> \u0993 Q<\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09af\u09bc \u09af\u09a5\u09be\u0995\u09cd\u09b0\u09ae\u09c7 z_{1}=r_{1}\\left(\\cos \\theta_{1}+i \\sin \\theta_{1}\\right)<\/span><\/span>\u00a0\u098f\u09ac\u0982 z_{2}=r_{2}\\left(\\cos \\theta_{2}+i \\sin \\theta_{2}\\right)<\/span><\/span>\u00a0\u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7 \u098f\u09ac\u0982 z_1<\/span> \u0993 z_2<\/span> \u098f\u09b0 \u09ad\u09be\u0997\u09ab\u09b2 =\\frac{z_1}{z_2}=z<\/span> \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6\u09bf\u09a4 \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 R<\/span> \u09a4\u09be\u09b9\u09b2\u09c7 O P=r_{1}=\\left|z_{1}\\right|, O Q=r_{2}=\\left|z_{2}\\right|, O R=\\frac{r_{1}}{r_{2}}=\\frac{\\left|z_{1}\\right|}{\\left|z_{2}\\right|}=\\frac{O P}{O Q}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\r\n
\"Complex<\/figure>\r\n<\/div>\r\n\r\n\r\n\\angle POX=\\theta_1,\\angle QOX=\\theta_2 \\;\u098f\u09ac\u0982\\; \\angle ROX=\\theta_2-\\theta_1<\/span>\n\r\n\r\n\r\n

OX<\/span><\/span> \u09ac\u09b0\u09be\u09ac\u09b0 OA=1<\/span><\/span> \u09ac\u09bf\u09ac\u09c7\u099a\u09a8\u09be \u0995\u09b0\u09bf \u098f\u09ac\u0982 OP<\/span><\/span> \u098f\u09b0 \u09af\u09c7 \u09aa\u09be\u09b6\u09c7 OQ<\/span><\/span> \u0986\u099b\u09c7, \u09a4\u09be\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09aa\u09be\u09b6\u09c7 \\angle QOX<\/span><\/span> \u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \\angle POR<\/span><\/span> \u0985\u0999\u09cd\u0995\u09a8 \u0995\u09b0\u09bf \u09af\u09c7\u09a8 \\angle QOP<\/span><\/span> \u098f\u09b0 \u09b8\u09ae\u09be\u09a8 \\angle OAR<\/span><\/span>\u00a0\u09b9\u09af\u09bc\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u099a\u09bf\u09a4\u09cd\u09b0\u09be\u09a8\u09c1\u09b8\u09be\u09b0\u09c7, <\/span><\/p>\r\n\r\n\r\n\\angle QOX=\\theta2=\\angle POR\\\\ \u09ac\u09be,\\; \\angle QOP+\\angle POX=\\angle POX+\\angle XOR\\\\ \u09ac\u09be,\\; \\angle QOP=\\angle XOR=\\angle AOR\\\\ \u0986\u09ac\u09be\u09b0,\\; \\angle OQP=\\angle OAR<\/span>\n\r\n\r\n\r\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 OPQ<\/span><\/span> \u098f\u09ac\u0982 AOR<\/span><\/span>\u00a0\u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c\u09a6\u09cd\u09ac\u09df \u09b8\u09a6\u09c3\u09b6\u0964<\/span><\/p>\r\n\r\n\r\n\r\n

\u09a4\u09be\u09b9\u09b2\u09c7, \\frac{OR}{OP}=\\frac{AR}{PQ}=\\frac{OA}{OQ}<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u09ac\u09be, OR=\\frac{OP}{OQ}.\u00a0OA=\\frac{OP}{OQ} [\\because OA=1]<\/span>\u00a0<\/span><\/p>\r\n\r\n\r\n\r\n

\u0986\u09ac\u09be\u09b0, \\angle XOR=\\angle POR-\\angle POX=\\theta_2-\\theta_1<\/span><\/span><\/p>\r\n\r\n\r\n\r\n

\u0985\u09a4\u098f\u09ac, R<\/span><\/span> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0987 \u09a6\u09c1\u0987\u099f\u09bf \u099c\u099f\u09bf\u09b2 \u09b8\u0982\u0996\u09cd\u09af\u09be z_1<\/span><\/span> \u0993 z_2<\/span><\/span> \u098f\u09b0 \u09ad\u09be\u0997\u09ab\u09b2 \\frac{z_1}{z_2}<\/span><\/span>\u00a0\u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09c7\u0964<\/span><\/p>\r\n

\r\n

 <\/p>\r\n

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