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If r satisfies the equation $r \times \left( {i + 2j + k} \right) = i - k$, then for any scalar $\alpha $, r is equal to
\[
  A.{\text{ }}i + \alpha \left( {i + 2j + k} \right) \\
  B.{\text{ j}} + \alpha \left( {i + 2j + k} \right) \\
  C.{\text{ k}} + \alpha \left( {i + 2j + k} \right) \\
  D.{\text{ }}i - k + \alpha \left( {i + 2j + k} \right) \\
\]

Answer Verified Verified
Hint: In this question assume $r = \alpha i + \beta j + \lambda k$ and then apply a cross product to reach the solution of the problem.

Complete step-by-step answer:
Let $r = \alpha i + \beta j + \lambda k$…………….. (1)
Now given equation is
$r \times \left( {i + 2j + k} \right) = i - k$
So, first find out the value of
$r \times \left( {i + 2j + k} \right)$
$ \Rightarrow r \times \left( {i + 2j + k} \right) = \left( {\alpha i + \beta j + \lambda k} \right) \times \left( {i + 2j + k} \right)$
Now, apply cross product property we have,
$ \Rightarrow r \times \left( {i + 2j + k} \right) = \left| {\begin{array}{*{20}{c}}
  i&j&k \\
  \alpha &\beta &\lambda \\
  1&2&1
\end{array}} \right| = i\left( {\beta - 2\lambda } \right) - j\left( {\alpha - \lambda } \right) + k\left( {2\alpha - \beta } \right)$
Now the above value is equal to
$i\left( {\beta - 2\lambda } \right) - j\left( {\alpha - \lambda } \right) + k\left( {2\alpha - \beta } \right) = i - k$
So on comparing the terms $i,{\text{ j, k}}$ we have
$\left( {\beta - 2\lambda } \right)$ = 1, $\left( {\alpha - \lambda } \right)$ = 0, $\left( {2\alpha - \beta } \right)$ = -1
$ \Rightarrow \beta = 1 + 2\lambda ,{\text{ }}\alpha = \lambda ,{\text{ }}\beta = 1 + 2\alpha $
So, from equation (1) we have,
$r = \alpha i + \beta j + \lambda k$, substitute the value of $\beta $ and $\lambda $ in this equation we have,
$ \Rightarrow r = \alpha i + \left( {1 + 2\alpha } \right)j + \alpha k$
$ \Rightarrow r = j + \alpha \left( {i + 2j + k} \right)$, for any scalar $\alpha $.
Hence, option (b) is correct.

Note: In such types of questions first assume r as above then apply cross product as above, then compare the value of cross product with the given value and find out the values of $\alpha ,{\text{ }}\beta {\text{, }}\lambda $, then substitute the value of $\beta $ and $\lambda $in terms of $\alpha $ in r we will get the required r for any scaler $\alpha $.
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