{"id":29,"date":"2022-03-24T18:05:29","date_gmt":"2022-03-24T18:05:29","guid":{"rendered":"https:\/\/stage-wp.10minuteschool.com\/?p=29"},"modified":"2023-06-19T16:36:38","modified_gmt":"2023-06-19T10:36:38","slug":"vector-algebra-formulas","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/vector-algebra-formulas\/","title":{"rendered":"\u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09ac\u09c0\u099c\u0997\u09be\u09a3\u09bf\u09a4\u09bf\u0995 \u09b8\u09c2\u09a4\u09cd\u09b0 (Vector Algebra Formulas)"},"content":{"rendered":"\r\n

\u00a0\u09e7. \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u09b8\u09c2\u09a4\u09cd\u09b0 (Commutative Law)<\/strong><\/h2>\r\n\r\n\r\n\r\n
\r\n
\"Commutative<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}}=\\overrightarrow{\\mathrm{B}}+\\overrightarrow{\\mathrm{A}}<\/span>\r\n\r\n\r\n\r\n

\u09a7\u09b0\u09be \u09af\u09be\u0995, \\overrightarrow{\\mathrm{OP}}=\\overrightarrow{\\mathrm{A}}<\/span> \u098f\u09ac\u0982 \\overrightarrow{\\mathrm{OR}}=\\overrightarrow{\\mathrm{B}}<\/span> \u09a6\u09c1\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 O \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09c7\u0964 \\mathrm{OPQR}<\/span>\u00a0 \u09b8\u09be\u09ae\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09aa\u09c2\u09b0\u09cd\u09a8\u00a0 \u0995\u09b0\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987,<\/p>\r\n\r\n\r\n\r\n\\begin{aligned}\n& \\overrightarrow{\\mathrm{OP}}+\\overrightarrow{\\mathrm{PQ}}=\\overrightarrow{\\mathrm{OQ}} \\\\\n& \\overrightarrow{\\mathrm{OR}}+\\overrightarrow{\\mathrm{RQ}}=\\overrightarrow{\\mathrm{OQ}} \\\\\n\\therefore & \\overrightarrow{\\mathrm{OP}}+\\overrightarrow{\\mathrm{PQ}}=\\overrightarrow{\\mathrm{OR}}+\\overrightarrow{\\mathrm{RQ}} \\\\\n\\Rightarrow & \\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}}=\\overrightarrow{\\mathrm{B}}+\\overrightarrow{\\mathrm{A}}\n\\end{aligned}<\/span>\r\n\r\n\r\n\r\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 (Vector Addition) \u09af\u09cb\u0997 \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u09b8\u09c2\u09a4\u09cd\u09b0 \u09ae\u09c7\u09a8\u09c7 \u099a\u09b2\u09c7\u0964<\/p>\r\n\r\n\r\n\r\n

\u09e8. \u09b8\u0982\u09af\u09cb\u0997\u09b8\u09c2\u09a4\u09cd\u09b0 (Associative Law)<\/strong><\/h2>\r\n\r\n\r\n\r\n(\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}})+\\overrightarrow{\\mathrm{C}}=\\overrightarrow{\\mathrm{A}}+(\\overrightarrow{\\mathrm{B}}+\\overrightarrow{\\mathrm{C}})<\/span>\r\n\r\n\r\n\r\n

\u09a7\u09b0\u09be \u09af\u09be\u0995, \\overrightarrow{\\mathrm{OP}}=\\overrightarrow{\\mathrm{A}} \\overrightarrow{\\mathrm{PQ}}=\\overrightarrow{\\mathrm{B}}<\/span> \u098f\u09ac\u0982 \\overrightarrow{\\mathrm{QR}}=\\overrightarrow{\\mathrm{C}}<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8 \\overrightarrow{\\mathrm{OP}}<\/span> \u098f\u09ac\u0982 \\overrightarrow{\\mathrm{PQ}}<\/span> \u09af\u09cb\u0997 \u0995\u09b0\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987,<\/p>\r\n\r\n\r\n\r\n\\begin{array}{l}\n\\overrightarrow{\\mathrm{OP}}+\\overrightarrow{\\mathrm{PQ}}=\\overrightarrow{\\mathrm{OQ}}=(\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}}) \\\\\n\\text { \u098f\u09ac\u0982 } \\overrightarrow{\\mathrm{PQ}}+\\overrightarrow{\\mathrm{QR}}=\\overrightarrow{\\mathrm{PR}}=(\\overrightarrow{\\mathrm{B}}+\\overrightarrow{\\mathrm{C}}) \\\\\n\\text { \u098f\u0996\u09a8 , } \\overrightarrow{\\mathrm{OQ}}+\\overrightarrow{\\mathrm{QR}}=\\overrightarrow{\\mathrm{OR}} \\\\\n\\text { \u0985\u09b0\u09cd\u09a5\u09be\u09ce, }(\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}})+\\overrightarrow{\\mathrm{C}}=\\overrightarrow{\\mathrm{D}} \\\\\n\\text { \u0986\u09ac\u09be\u09b0, } \\overrightarrow{\\mathrm{OP}}+\\overrightarrow{\\mathrm{PR}}=\\overrightarrow{\\mathrm{OR}} \\\\\n\\text { \u0985\u09b0\u09cd\u09a5\u09be\u09ce } \\overrightarrow{\\mathrm{A}}+(\\overrightarrow{\\mathrm{B}}+\\overrightarrow{\\mathrm{C}})=\\overrightarrow{\\mathrm{D}} \\\\\n(\\vec{A}+\\vec{B})+\\vec{C}=\\vec{A}+(\\vec{B}+\\vec{C})\n\\end{array}<\/span>\r\n\r\n\r\n\r\n

\u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09af\u09cb\u0997 (Vector Addition) \u09b8\u0982\u09af\u09cb\u0997 \u09b8\u09c2\u09a4\u09cd\u09b0 \u09ae\u09c7\u09a8\u09c7 \u099a\u09b2\u09c7\u0964 \u00a0\u0985\u09a4\u098f\u09ac, \u09a6\u09c7\u0996\u09be \u09af\u09be\u09af\u09bc \u09af\u09c7 \u09ac\u09b9\u09c1\u09b2\u0982\u0996\u09cd\u09af\u0995 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09b2\u09ac\u09cd\u09a7\u09bf \u09a4\u09be\u09a6\u09c7\u09b0 \u09af\u09cb\u0997\u09c7\u09b0 \u0995\u09cd\u09b0\u09ae\u09c7\u09b0 \u0989\u09aa\u09b0 \u09a8\u09bf\u09b0\u09cd\u09ad\u09b0 \u0995\u09b0\u09c7 \u09a8\u09be\u0964<\/p>\r\n\r\n\r\n\r\n

\u09e9. \u09ac\u09a3\u09cd\u099f\u09a8\u09b8\u09c2\u09a4\u09cd\u09b0 (Distributive Law)<\/strong><\/h2>\r\n\r\n\r\n\r\n
\r\n
\"Distributive<\/figure>\r\n<\/div>\r\n\r\n\r\n\r\n\\quad \\mathrm{m}(\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}})=\\mathrm{m} \\overrightarrow{\\mathrm{A}}+\\mathrm{m} \\overrightarrow{\\mathrm{B}}<\/span>\r\n\r\n\r\n\r\n

\u09a7\u09b0\u09be \u09af\u09be\u0995, \\overrightarrow{\\mathrm{OP}}=\\overrightarrow{\\mathrm{A}}<\/span> \u098f\u09ac\u0982 \\overrightarrow{\\mathrm{PR}}=\\overrightarrow{\\mathrm{B}}<\/span><\/p>\r\n\r\n\r\n\r\n

\u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09af\u09cb\u0997\u09c7\u09b0 (Vector Addition) \u09a8\u09bf\u09af\u09bc\u09ae \u0985\u09a8\u09c1\u09b8\u09be\u09b0\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, \\begin{aligned}\n\\overrightarrow{\\mathrm{OR}} &=\\overrightarrow{\\mathrm{OP}}+\\overrightarrow{\\mathrm{PR}} \\\\\n&=\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}}\n\\end{aligned}<\/span><\/p>\r\n\r\n\r\n\r\n

\u098f\u0996\u09a8 \u09a7\u09b0\u09be \u09af\u09be\u0995, 0P\u00a0 \u098f\u09ac\u0982 OR \u098f\u09b0 \u09ac\u09b0\u09cd\u09a7\u09bf\u09a4\u09be\u0982\u09b6\u09c7\u09b0 \u0989\u09aa\u09b0 Q\u00a0 \u098f\u09ac\u0982 S \u00a0\u09a6\u09c1\u099f\u09bf \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a8\u09c7\u09af\u09bc\u09be \u09b9\u09af\u09bc \u09af\u09be\u09a4\u09c7, \\begin{aligned} & \\overrightarrow{\\mathrm{OQ}}=\\mathrm{m} \\cdot \\overrightarrow{\\mathrm{OP}}=\\mathrm{m} \\overrightarrow{\\mathrm{A}} \\\\ \\text { \u098f\u09ac\u0982 } & \\overrightarrow{\\mathrm{QS}}=\\mathrm{m} \\cdot \\overrightarrow{\\mathrm{PR}}=\\mathrm{m} \\overrightarrow{\\mathrm{B}} \\text { \u09b9\u09af\u09bc\u0964 } \\\\ \\therefore & \\frac{\\mathrm{OQ}}{\\mathrm{OP}}=\\frac{\\mathrm{QS}}{\\mathrm{PR}}=\\frac{\\mathrm{OS}}{\\mathrm{OR}}=\\mathrm{m} \\\\ \\therefore & \\overrightarrow{\\mathrm{OS}}=\\mathrm{m} \\cdot \\overrightarrow{\\mathrm{OR}} \\\\ \\Rightarrow & \\overrightarrow{\\mathrm{OS}}=\\mathrm{m}(\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}}) \\quad[\\because \\overrightarrow{\\mathrm{OR}}=\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}}] \\end{aligned}<\/span><\/p>\r\n\r\n\r\n\r\n

\u0986\u09ac\u09be\u09b0, \u00a0 \\overrightarrow{\\mathrm{OS}}=\\overrightarrow{\\mathrm{OQ}}+\\overrightarrow{\\mathrm{QS}}<\/span><\/p>\r\n\r\n\r\n\r\n\\begin{array}{c}\n=\\mathrm{m} \\overrightarrow{\\mathrm{A}}+\\mathrm{m} \\overrightarrow{\\mathrm{B}} \\\\\n\\therefore \\mathrm{m}(\\overrightarrow{\\mathrm{A}}+\\overrightarrow{\\mathrm{B}})=\\mathrm{m} \\overrightarrow{\\mathrm{A}}+\\mathrm{m} \\overrightarrow{\\mathrm{B}}\n\\end{array}<\/span>\r\n","protected":false},"excerpt":{"rendered":"

\u00a0\u09e7. \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u09b8\u09c2\u09a4\u09cd\u09b0 (Commutative Law) \u09a7\u09b0\u09be \u09af\u09be\u0995, \u098f\u09ac\u0982 \u09a6\u09c1\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 O \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a4\u09c7 \u0995\u09cd\u09b0\u09bf\u09af\u09bc\u09be \u0995\u09b0\u09c7\u0964 \u00a0 \u09b8\u09be\u09ae\u09a8\u09cd\u09a4\u09b0\u09bf\u0995 \u09aa\u09c2\u09b0\u09cd\u09a8\u00a0 \u0995\u09b0\u09c7 \u0986\u09ae\u09b0\u09be \u09aa\u09be\u0987, \u09b8\u09c1\u09a4\u09b0\u09be\u0982 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 (Vector Addition) \u09af\u09cb\u0997 \u09ac\u09bf\u09a8\u09bf\u09ae\u09af\u09bc \u09b8\u09c2\u09a4\u09cd\u09b0 \u09ae\u09c7\u09a8\u09c7 \u099a\u09b2\u09c7\u0964 \u09e8. \u09b8\u0982\u09af\u09cb\u0997\u09b8\u09c2\u09a4\u09cd\u09b0 (Associative Law) \u09a7\u09b0\u09be \u09af\u09be\u0995, \u098f\u09ac\u0982 \u098f\u0996\u09a8 \u098f\u09ac\u0982 \u09af\u09cb\u0997 \u0995\u09b0\u09c7<\/p>\n

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