LinAlg4.2.17

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If the two vectors {\bar  {a}}=2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}},{\bar  {b}}=5{\bar  {i}}-3{\bar  {j}}+{\bar  {k}}\, are two vectors,find the projection of {\bar  {b}}\, on {\bar  {a}}\,

Orthogonal projection of vector b on vector a is {\frac  {({\bar  {b}}\cdot {\bar  {a}}){\bar  {a}}}{|a|^{2}}}\,

This is equal to {\frac  {[(5{\bar  {i}}-3{\bar  {j}}+{\bar  {k}})\cdot (2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}})](2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}})}{{\sqrt  {4+1+4}}}}={\frac  {(10-3+2)(2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}})}{9}}=2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}}\,

Therefore the projection is 2{\bar  {i}}+{\bar  {j}}+2{\bar  {k}}\,


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