Equation of the plane passing through the points with position vectors (3,-5,-1), (-1,5,7) and parallel to the vector (3,-1,7) is

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$3 x + 2 y + z = 0$

$3 x + 2 y - z = 0$

$3 x - 2 y - z = 0$

$3 x + 2 y + 3 z = 0$

answer is B.

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Detailed Solution

$\begin{array}{rcl} |\begin{matrix} x - 3 & y + 5 & z + 1 \\ - 4 & 10 & 8 \\ 3 & - 1 & 7 \end{matrix}| & = & 0 \end{array}$

$\begin{array}{rcl} (x - 3) (70 + 8) - (y + 5) (- 28 - 24) + (z + 1) (4 - 30) & = & 0 \end{array}$

$\begin{array}{rcl} 6 (x - 3) + 4 (y + 5) - 2 (z + 1) & = & 0 \end{array}$

$\begin{array}{rcl} \Rightarrow & 3 x - 9 + 2 y + 10 - z - 1 = 0 \\ \Rightarrow & 3 x + 2 y - z = 0 \end{array}$

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