The equation of a line passing through the point $(- 1, 2, - 3)$ and perpendicular to the plane $2 x + 3 y + z + 5 = 0$ is

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$\frac{x + 1}{- 1} = \frac{y - 2}{1} = \frac{z + 3}{1}$

$\frac{x - 1}{- 1} = \frac{y + 2}{1} = \frac{z - 3}{- 1}$

$\frac{x + 1}{2} = \frac{y - 2}{3} = \frac{z + 3}{1}$

$\frac{x + 1}{- 2} = \frac{y - 2}{2} = \frac{z + 3}{- 1}$

answer is C.

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Detailed Solution

The given point is $(- 1, 2, - 3)$ and the given plane is $2 x + 3 y + z + 5 = 0$

We know that the equation of line passing through the point $(x_{1}, y_{1}, z_{1})$ and perpendicular to the plane $a x + b y + c z + d = 0$ is $\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$

Therefore, the equation of the line passing through the point $(- 1, 2, - 3)$ and perpendicualr to the plane $2 x + 3 y + z + 5 = 0$ is $\frac{x + 1}{2} = \frac{y - 2}{3} = \frac{z + 3}{1}$