The position vector of point of intersection of the planes r→.(n^2×n^3)=1,r→.(n^3×n^1)=2 and r→.(n^1×n^2)=3([n^1n^2n^3]≠0) is

r→=(r→·(n^2×n^3))n^1+(r→·(n^3×n^1))n^2+(r→·(n^1×n^2))n^3[n^1n^2n^3]=n^1+2n^2+3n^3[n^1n^2n^3]

The position vector of point of intersection of the planes r→.(n^2×n^3)=1,r→.(n^3×n^1)=2 and r→.(n^1×n^2)=3([n^1n^2n^3]≠0) is

r→=(r→·(n^2×n^3))n^1+(r→·(n^3×n^1))n^2+(r→·(n^1×n^2))n^3[n^1n^2n^3]=n^1+2n^2+3n^3[n^1n^2n^3]

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The position vector of point of intersection of the planes $\vec{r} . ({\hat{n}}_{2} \times {\hat{n}}_{3}) = 1, \vec{r} . ({\hat{n}}_{3} \times {\hat{n}}_{1}) = 2$ and $\vec{r} . ({\hat{n}}_{1} \times {\hat{n}}_{2}) = 3 ([{\hat{n}}_{1} {\hat{n}}_{2} {\hat{n}}_{3}] \neq 0)$ is

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$\frac{{\hat{n}}_{2} \times {\hat{n}}_{3} + 2 {\hat{n}}_{3} \times {\hat{n}}_{1} + 3 {\hat{n}}_{1} \times {\hat{n}}_{2}}{{[{\hat{n}}_{1} {\hat{n}}_{2} {\hat{n}}_{3}]}^{2}}$

$\frac{{\hat{n}}_{2} \times {\hat{n}}_{3} + 2 {\hat{n}}_{3} \times {\hat{n}}_{1} + 3 {\hat{n}}_{1} \times {\hat{n}}_{2}}{[{\hat{n}}_{1} {\hat{n}}_{2} {\hat{n}}_{3}]}$

$\frac{{\hat{n}}_{1} + 2 {\hat{n}}_{2} + 3 {\hat{n}}_{3}}{[{\hat{n}}_{1} {\hat{n}}_{2} {\hat{n}}_{3}]}$

$\frac{{\hat{n}}_{1} + 2 {\hat{n}}_{2} + 3 {\hat{n}}_{3}}{{[{\hat{n}}_{1} {\hat{n}}_{2} {\hat{n}}_{3}]}^{2}}$

answer is A.

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Detailed Solution

$\vec{r} = \frac{(\vec{r} \cdot ({\hat{n}}_{2} \times {\hat{n}}_{3})) {\hat{n}}_{1} + (\vec{r} \cdot ({\hat{n}}_{3} \times {\hat{n}}_{1})) {\hat{n}}_{2} + (\vec{r} \cdot ({\hat{n}}_{1} \times {\hat{n}}_{2})) {\hat{n}}_{3}}{[{\hat{n}}_{1} {\hat{n}}_{2} {\hat{n}}_{3}]}$