LinAlg4.1.11

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In a triangle ABC if A=2{\bar  {i}}+4{\bar  {j}}-{\bar  {k}},B=4{\bar  {i}}+5{\bar  {j}}+{\bar  {k}},C=3{\bar  {i}}+6{\bar  {j}}-3{\bar  {k}}\, and D is the mid point of the side BC, then find the length of AD.

Given D is the mid point of BC.

Therefore,Pposition vector of D={\frac  {1}{2}}(4{\bar  {i}}+5{\bar  {j}}+{\bar  {k}}+3{\bar  {i}}+6{\bar  {j}}-3{\bar  {k}})={\frac  {1}{2}}(7{\bar  {i}}+11{\bar  {j}}-2{\bar  {k}})\,

Therefore,{\bar  {AD}}=[{\frac  {7}{2}}{\bar  {i}}+{\frac  {11}{2}}{\bar  {j}}-{\bar  {k}}]-[2{\bar  {i}}-4{\bar  {j}}-{\bar  {k}}]={\frac  {3}{2}}{\bar  {i}}+{\frac  {3}{2}}{\bar  {j}}\,

Hence |AD|={\frac  {3}{2}}{\sqrt  {1+1}}={\frac  {3}{{\sqrt  {2}}}}\,


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