{"id":10659,"date":"2022-03-24T18:06:39","date_gmt":"2022-03-24T18:06:39","guid":{"rendered":"https:\/\/10minuteschool.com\/content\/?p=10659"},"modified":"2025-05-07T15:27:46","modified_gmt":"2025-05-07T09:27:46","slug":"addition-subtraction-scalar-multiplication-vectors","status":"publish","type":"post","link":"https:\/\/10minuteschool.com\/content\/addition-subtraction-scalar-multiplication-vectors\/","title":{"rendered":"\u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09af\u09cb\u0997, \u09ac\u09bf\u09df\u09cb\u0997 , \u09b8\u09cd\u0995\u09c7\u09b2\u09be\u09b0 \u0997\u09c1\u09a3\u09bf\u09a4\u0995 \u0993 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3(Addition, Subtraction, Scalar multiplication vectors)"},"content":{"rendered":"<h3 style=\"text-align: center;\"><\/h3>\n<h3><span style=\"color: #000000;\"><b>(i) \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09af\u09cb\u0997 <\/b><b>(<\/b><b>Addition of vectors)<\/b><b> :<\/b><\/span><\/h3>\n<p>a \u0993 b \u09a6\u09c1\u0987\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0964 b\u00a0 \u098f\u09b0 \u0986\u09a6\u09bf\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 a \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09b2\u09c7, a \u098f\u09b0 \u0986\u09a6\u09bf\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 b \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u0982\u09af\u09cb\u099c\u0995 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u0982\u09b6 \u09b9\u09b2\u09cb \u09ad\u09c7\u0995\u09cd\u099f\u09b0 a\u00a0 \u0993 b \u098f\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u098f\u0995\u09c7 a+b\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-10660 size-medium\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture1-1-300x177.png\" alt=\"\u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09af\u09cb\u0997\" width=\"300\" height=\"177\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture1-1-300x177.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture1-1.png 430w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h3><b>(ii) \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09ac\u09bf\u09df\u09cb\u0997 <\/b><b>(<\/b><b>Subtraction of vectors)<\/b><b>:<\/b><b>\u00a0<\/b><\/h3>\n<p>a \u0993 b \u09a6\u09c1\u0987\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0964 b\u00a0 \u098f\u09b0 \u0986\u09a6\u09bf\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 a \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09b2\u09c7, b \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 a \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u0982\u09af\u09cb\u099c\u0995 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u0982\u09b6 \u09b9\u09b2\u09cb \u09ad\u09c7\u0995\u09cd\u099f\u09b0 a\u00a0 \u0993 b \u098f\u09b0 \u09ac\u09bf\u09df\u09cb\u0997\u09ab\u09b2 \u098f\u0995\u09c7 a-b\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-10661 size-medium\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture2-1-300x173.png\" alt=\"Subtraction of vectors\" width=\"300\" height=\"173\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture2-1-300x173.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture2-1.png 436w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p><b>(iii) \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09b8\u09cd\u0995\u09c7\u09b2\u09be\u09b0 \u0997\u09c1\u09a3\u09bf\u09a4\u0995 <\/b><b>(<\/b><b>Sclar multiple of vectors)<\/b><b>:\u00a0<\/b><\/p>\n<p>\u0995\u09cb\u09a8\u09cb \u09ad\u09c7\u0995\u09cd\u099f\u09b0 a \u0995\u09c7 \u098f\u0995\u099f\u09bf \u09b8\u09cd\u0995\u09c7\u09b2\u09be\u09b0 \u0997\u09c1\u09a8\u09bf\u09a4\u0995 m \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0997\u09c1\u09a8 \u0995\u09b0\u09b2\u09c7 \u0997\u09c1\u09a8\u09ab\u09b2 ma \u0993 \u098f\u0995\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b9\u09df\u0964<\/p>\n<p>ma<b> \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09ac\u09c8\u09b6\u09bf\u09b7\u09cd\u099f\u09cd\u09af :-\u00a0<\/b><\/p>\n<p>(i) ma \u098f\u09b0 \u09ae\u09be\u09a8 =ma=ma\u00a0;\u00a0 \u0985\u09b0\u09cd\u09a5\u09be\u09ce \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u099f\u09bf\u09b0 \u09a6\u09c8\u09b0\u09cd\u0998\u09cd\u09af \u09b9\u09ac\u09c7 a \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09ae\u09be\u09a8\u09c7\u09b0 m \u0997\u09c1\u09a8\u0964<\/p>\n<p>(ii) ma \u098f\u09b0 \u09a6\u09bf\u0995 a \u098f\u09b0 \u09a6\u09bf\u0995\u00a0 \u098f\u0995\u0987 \u09b9\u09ac\u09c7 \u09af\u09a6\u09bf m \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09df ma \u098f\u09b0 \u09a6\u09bf\u0995 a \u098f\u09b0 \u09a6\u09bf\u0995\u09c7\u09b0 \u09ac\u09bf\u09aa\u09b0\u09c0\u09a4 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 \u09af\u09a6\u09bf m \u098b\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b9\u09df \u0964<\/p>\n<p>(iii) \u09af\u09a6\u09bf m=0 \u09b9\u09df \u09a4\u09ac\u09c7 ma=0 \u09b9\u09ac\u09c7 \u0964<\/p>\n<p>(iv) ma \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u098f\u09ac\u0982 a \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09a7\u09a8\u09be\u09a4\u09cd\u09ae\u0995 \u09b0\u09c7\u0996\u09be \u098f\u0995\u0987 \u09ac\u09be \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b9\u09df\u0964<\/p>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center;\"><b>\u09a4\u09cd\u09b0\u09bf\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u099c\u0997\u09a4\u09c7 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u0993 \u09b8\u09cd\u0995\u09c7\u09b2\u09be\u09b0 \u0997\u09c1\u09a8\u09bf\u09a4\u0995\u0995\u09c7 <\/b>i\u00a0,\u00a0j\u00a0,\u00a0k<b> \u098f\u09b0 \u09ae\u09be\u09a7\u09cd\u09af\u09ae\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 <\/b><b>(Representation of vector addition and scalar multiple on terms of <\/b>i\u00a0,\u00a0j\u00a0,\u00a0k<b> in three dimensional space)<\/b><\/h2>\n<p>\u09a7\u09b0\u09bf, a=a1i+a2j+a3k<\/p>\n<p>b=b1i+b2j+b3k<\/p>\n<p>a \u0993 b \u098f\u09b0 \u09b2\u09a6\u09cd\u09a7\u09bf , a+b=a1+b1 i+a2+b2j+a3+b3 k<\/p>\n<p>\u098f\u09ac\u0982 a \u098f\u09b0 m \u0997\u09c1\u09a3\u09bf\u09a4\u0995 , ma=ma1i+a2j+a3k<\/p>\n<p>=ma1i+ma2j+ma3k<\/p>\n<h2 style=\"text-align: center;\"><b>\u09b8\u09ae\u09a4\u09b2\u09c7 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u0985\u0982\u09b6\u0995 <\/b><b>(Components of vector in a plane)<br \/>\n<\/b><\/h2>\n<p>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, a \u098f\u0995\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09af\u09be\u09b0 \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 (0,0)\u00a0 \u098f\u09ac\u0982 \u09b6\u09c0\u09b0\u09cd\u09b7\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 P(ax,ay) \u09a4\u09be\u09b9\u09b2\u09c7 OP=a \u09b9\u09ac\u09c7 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0964<\/p>\n<p>\u0986\u09ae\u09b0\u09be \u099c\u09be\u09a8\u09bf, \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09ae\u09be\u09a8\u09c7\u09b0 \u09b8\u09be\u09a5\u09c7 \u0990 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u099f\u09bf \u098f\u0995\u0995 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b0\u09be\u09b6\u09bf \u09b9\u09df\u0964 X- \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u0995 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 i \u09a6\u09cd\u09ac\u09be\u09b0\u09be ax \u0995\u09c7 \u0997\u09c1\u09a3 \u0995\u09b0\u09b2\u09c7 ON=ax\u00a0i \u098f\u0995\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b0\u09be\u09b6\u09bf \u09b9\u09ac\u09c7 \u09af\u09be\u09b0 \u09ae\u09be\u09a8 \u09b9\u09ac\u09c7 ax \u098f\u09ac\u0982 \u09a6\u09bf\u0995 \u09b9\u09ac\u09c7 X- \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7\u0964<\/p>\n<p>\u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7 NP=ay\u00a0j \u098f\u0995\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09af\u09be\u09b0 \u09ae\u09be\u09a8 \u09b9\u09b2\u09cb ay \u098f\u09ac\u0982 \u09a6\u09bf\u0995 \u09b9\u09b2\u09cb Y- \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u098f\u0996\u09be\u09a8\u09c7 j Y- \u0985\u0995\u09cd\u09b7\u09c7\u09b0 \u09a6\u09bf\u0995\u09c7 \u098f\u0995\u099f\u09bf \u098f\u0995\u0995 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0964<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-10662 size-medium\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture3-1-300x277.png\" alt=\"Components of vector in a plane\" width=\"300\" height=\"277\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture3-1-300x277.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture3-1.png 406w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>\u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09af\u09cb\u0997\u09c7\u09b0 \u09a4\u09cd\u09b0\u09bf\u09ad\u09c1\u099c \u09b8\u09c2\u09a4\u09cd\u09b0 \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0987,<\/p>\n<p>OP=ON+NP<\/p>\n<p>\u09ac\u09be, a=ax+ay<\/p>\n<p>axi \u098f\u09ac\u0982 ay j \u09b9\u09b2\u09c7 a \u098f\u09b0 \u0989\u09aa\u09be\u0982\u09b6\u0964<\/p>\n<p>a \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09ae\u09be\u09a8 a=ax2+ay2<\/p>\n<p>&nbsp;<\/p>\n<h2 style=\"text-align: center;\"><b>\u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995\u09c7 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 <\/b><b>(Representaltion of vector in cartesian coordinates)<br \/>\n<\/b><\/h2>\n<p>\u0986\u09df\u09a4\u0995\u09be\u09b0 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 Y- \u09ac\u09cd\u09af\u09ac\u09b8\u09cd\u09a5\u09be\u09df X,Y,Z \u0985\u0995\u09cd\u09b7 \u09ac\u09b0\u09be\u09b0\u09b0 \u098f\u0995\u0995 \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0995\u09c7 i,j,k \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09a8\u09bf\u09b0\u09cd\u09a6\u09c7\u09b6 \u0995\u09b0\u09c7\u0964<\/p>\n<p>\u099a\u09bf\u09a4\u09cd\u09b0\u09c7 P\u00a0\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u09cd\u09a5\u09be\u09a8\u09be\u0999\u09cd\u0995 (x,y)\u00a0\u09b9\u09b2\u09c7, OP=xi+yj \u098f\u0995\u09c7 OP=x y \u00a0\u09ac\u09be, OP=(x,y) \u09a6\u09cd\u09ac\u09be\u09b0\u09be \u0993 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09b9\u09df\u0964<\/p>\n<p>\u0985\u09a8\u09c1\u09b0\u09c2\u09aa\u09ad\u09be\u09ac\u09c7, \u09a4\u09cd\u09b0\u09bf\u09ae\u09be\u09a4\u09cd\u09b0\u09bf\u0995 \u099c\u0997\u09a4\u09c7 xi+yj+zk\u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0995\u09c7 x y z \u09ac\u09be, x,y,z \u09a6\u09cd\u09ac\u09be\u09b0\u09be\u0993 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09df\u0964<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-10663 size-medium\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture4-1-300x211.png\" alt=\"Representaltion of vector in cartesian coordinates\" width=\"300\" height=\"211\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture4-1-300x211.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture4-1.png 435w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2 style=\"text-align: center;\"><b>\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b><b>(Vector equation of straight line)<\/b><\/h2>\n<p><b>(i) \u09a6\u09c7\u0996\u09be\u0993 \u09af\u09c7, <\/b>A(a)<b> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u098f\u09ac\u0982 <\/b>b<b> \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u098f\u09ac\u0982 <\/b>b\u00a0<b>\u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 <\/b>r=a+tb<b>\u00a0 \u09af\u09c7\u0996\u09be\u09a8\u09c7 <\/b>t<b> \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09af\u09be\u09b0\u09be\u09ae\u09bf\u099f\u09be\u09b0 \u098f\u09ac\u0982 \u0986\u09b0\u09cb \u09a6\u09c7\u0996\u09be\u0993 \u09af\u09c7, \u098f\u0995\u09c7 <\/b>r-a\u00d7b=0<b> \u0986\u0995\u09be\u09b0\u09c7\u0993 \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be \u09af\u09be\u09df\u0964<\/b><\/p>\n<p><b>\u09aa\u09cd\u09b0\u09ae\u09be\u09a3 :<\/b><\/p>\n<p>\u09ae\u09a8\u09c7 \u0995\u09b0\u09bf, \u09ae\u09c2\u09b2\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 O \u098f\u09ac\u0982 OA=a\u0964 \u09a7\u09b0\u09bf , A(a) \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u098f\u09ac\u0982 b \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2<\/p>\n<p>\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u0989\u09aa\u09b0 P \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09af\u09be\u09b0 \u0985\u09ac\u09b8\u09cd\u09a5\u09be\u09a8 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 OP=r<\/p>\n<p>\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 AP \u098f\u09ac\u0982 b \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2, \u2234AP=tb \u09af\u09c7\u0996\u09be\u09a8\u09c7 t \u098f\u0995\u099f\u09bf \u09aa\u09cd\u09af\u09be\u09b0\u09be\u09ae\u09bf\u099f\u09be\u09b0<\/p>\n<p>OA+AP=OP \u09ac\u09be, a+tb=r \u09ac\u09be, r=a+tb<\/p>\n<p>\u0986\u09ac\u09be\u09b0, AP \u098f\u09ac\u0982 b \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09ac\u09b2\u09c7 AP\u00d7b=0 \u09ac\u09be, r-a\u00d7b=0<\/p>\n<p><b>(ii) <\/b>A(a)<b> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u0997\u09be\u09ae\u09c0 \u098f\u09ac\u0982 <\/b>B(b)<b> \u0993 <\/b>C(c)<b> \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09a6\u09cd\u09ac\u09df\u09c7\u09b0 \u09b8\u0982\u09af\u09cb\u0997 \u09b8\u09b0\u09b2\u09b0\u09b0\u09c7\u0996\u09be\u09b0 \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a8\u09bf\u09b0\u09cd\u09a3\u09df \u0995\u09b0\u0964\u00a0\u00a0<\/b><\/p>\n<p>\u27f9 P(r) \u09af\u09c7\u0995\u09cb\u09a8\u09cb \u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b9\u09b2\u09c7,<\/p>\n<p>\u2234\u00a0AP=r-a\u00a0\u00a0\u00a0\u2026\u2026\u2026(1)<\/p>\n<p>\u0986\u09ac\u09be\u09b0, BC=c-b\u00a0\u00a0\u2026\u2026\u2026(2)<\/p>\n<p>\u09af\u09c7\u09b9\u09c7\u09a4\u09c1 AP \u098f\u09ac\u0982 BC \u09b8\u09ae\u09be\u09a8\u09cd\u09a4\u09b0\u09be\u09b2<\/p>\n<p>\u2234\u00a0AP=t BC<\/p>\n<p>\u09ac\u09be, r-a=t(c-b)<\/p>\n<p>\u09ac\u09be, r=a+t(c-b)<\/p>\n<p>&nbsp;<\/p>\n<p><img loading=\"lazy\" class=\"alignnone wp-image-10666 size-medium\" src=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture6-300x89.png\" alt=\"\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3\" width=\"300\" height=\"89\" srcset=\"https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture6-300x89.png 300w, https:\/\/10minuteschool.com\/content\/wp-content\/uploads\/2022\/03\/Picture6.png 410w\" sizes=\"(max-width: 300px) 100vw, 300px\" \/><\/p>\n<h2 style=\"text-align: center;\"><b>\u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u09b0 \u09ad\u09c7\u0995\u09cd\u099f\u09b0 \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a5\u09c7\u0995\u09c7 \u0995\u09be\u09b0\u09cd\u09a4\u09c7\u09b8\u09c0\u09df \u09b8\u09ae\u09c0\u0995\u09b0\u09a3<br \/>\n<\/b><\/h2>\n<p>a=a1i+a2j+a3k , b=b1i+b2j+b3k\u00a0,\u00a0r=xi+yj+zk \u09b9\u09b2\u09c7 r=a+tb \u09b8\u09ae\u09c0\u0995\u09b0\u09a3 \u09a5\u09c7\u0995\u09c7 \u09aa\u09be\u0993\u09df\u09be \u09af\u09be\u09df,<\/p>\n<p>xi+yj+zk=a1i+a2j+a3k+b1i+b2j+b3k<\/p>\n<p>\u09ac\u09be, x-a1i+y-a2j+z-a3k=t(b1i+b2j+b3k)<\/p>\n<p>\u0989\u09ad\u09df\u09aa\u0995\u09cd\u09b7 \u09a5\u09c7\u0995\u09c7 i,\u00a0\u00a0j\u00a0,\u00a0k \u098f\u09b0 \u09b8\u09b9\u0997 \u09b8\u09ae\u09c0\u0995\u09c3\u09a4 \u0995\u09b0\u09c7 \u09aa\u09be\u0987,<\/p>\n<p>\u09ac\u09be, x-a1i=tb1,\u00a0y-a2=tb2z-a3= tb3<\/p>\n<p>\u09ac\u09be, x-a1b1=y-a2b2=z-a3b3=t<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>(i) \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u09c7\u09b0 \u09af\u09cb\u0997 (Addition of vectors) : a \u0993 b \u09a6\u09c1\u0987\u099f\u09bf \u09ad\u09c7\u0995\u09cd\u099f\u09b0\u0964 b\u00a0 \u098f\u09b0 \u0986\u09a6\u09bf\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 a \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09b8\u09cd\u09a5\u09be\u09aa\u09a8 \u0995\u09b0\u09b2\u09c7, a \u098f\u09b0 \u0986\u09a6\u09bf\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1 \u09a5\u09c7\u0995\u09c7 b \u098f\u09b0 \u09aa\u09cd\u09b0\u09be\u09a8\u09cd\u09a4\u09ac\u09bf\u09a8\u09cd\u09a6\u09c1\u09b0 \u09b8\u0982\u09af\u09cb\u099c\u0995 \u09b8\u09b0\u09b2\u09b0\u09c7\u0996\u09be\u0982\u09b6 \u09b9\u09b2\u09cb \u09ad\u09c7\u0995\u09cd\u099f\u09b0 a\u00a0 \u0993 b \u098f\u09b0 \u09af\u09cb\u0997\u09ab\u09b2 \u098f\u0995\u09c7 a+b\u00a0\u09a6\u09cd\u09ac\u09be\u09b0\u09be \u09aa\u09cd\u09b0\u0995\u09be\u09b6 \u0995\u09b0\u09be<\/p>\n<p> <a class=\"redmore\" href=\"https:\/\/10minuteschool.com\/content\/addition-subtraction-scalar-multiplication-vectors\/\">Read More<\/a><\/p>\n","protected":false},"author":36,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":[],"categories":[4236,3028,50,51],"tags":[3904,3902,3901,3903,3905],"_links":{"self":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/10659"}],"collection":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/users\/36"}],"replies":[{"embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/comments?post=10659"}],"version-history":[{"count":9,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/10659\/revisions"}],"predecessor-version":[{"id":16178,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/posts\/10659\/revisions\/16178"}],"wp:attachment":[{"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/media?parent=10659"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/categories?post=10659"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/10minuteschool.com\/content\/wp-json\/wp\/v2\/tags?post=10659"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}