LinAlg4.2.14

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Let {\bar  {a}}=2{\bar  {i}}+3{\bar  {j}}+4{\bar  {k}},{\bar  {b}}={\bar  {i}}-2{\bar  {j}}+{\bar  {k}}\,.If a vector {\bar  {r}}\, satisfies {\bar  {a}}\times {\bar  {r}}=3{\bar  {b}}\, and {\bar  {a}}\cdot {\bar  {r}}=2\, then find the vector {\bar  {r}}\,

Given {\bar  {a}}=2{\bar  {i}}+3{\bar  {j}}+4{\bar  {k}},{\bar  {b}}={\bar  {i}}-2{\bar  {j}}+{\bar  {k}}\,.Let {\bar  {r}}=x{\bar  {i}}+y{\bar  {j}}+z{\bar  {k}}\,

Also given {\bar  {a}}\times {\bar  {r}}=3{\bar  {b}}={\begin{vmatrix}{\bar  {i}}&{\bar  {j}}&{\bar  {k}}\\2&3&4\\x&y&z\end{vmatrix}}=3({\bar  {i}}-2{\bar  {j}}+{\bar  {k}})\,

Hence {\bar  {i}}(3z-4y)-{\bar  {j}}(2z-4x)+{\bar  {k}}(2y-3x)=3{\bar  {i}}-6{\bar  {j}}+3{\bar  {k}}\,

3z-4y=3,4x-2z=-6,2y-3x=3\,

Given that {\bar  {a}}\cdot {\bar  {r}}=2,2x+3y+4z=2\,

2x=z-3,4y=3z-3\, Solving we get

z=1,x=-1,y=0\,

Hence {\bar  {r}}=-{\bar  {i}}+{\bar  {k}}\,


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