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The vector equation of the plane passing through $i + j + k$ and parallel to the vectors $2 i + 3 j - k$ , $i + 2 j + 3 k$ is

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$[\begin{array}{lll} r - (i + j + k) 2 i + 3 j - k i + 2 j + 3 k \end{array}] = 0$

$[\begin{array}{lll} r 2 i + 3 j - k i + 2 j + 5 k \end{array}] = 0$

$[\begin{array}{ccc} r - (i + j + k) & i + 2 j - 2 k & j + 2 k \end{array}] = 0$

$[\begin{array}{ccc} r & i + 2 j - 2 k & j + 2 k \end{array}] = 0$

answer is A.

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Detailed Solution

$V = [\begin{array}{lll} \bar{r} - \bar{a} \bar{b} \bar{c} \end{array}] = 0$

$Let a = i + j + k, b = 2 i + 3 j - k, c = i + 2 j + 3 k$

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