Find an equation of the plane that passes through the point \( P_{0}(-3,-2,3) \) with a normal vector \( n=\langle-1,2,-3\rangle \). Which of the following equations is an equation of the plane that passes through the point \( P_{0}(-3,-2,3) \) with a normal vector \( n=\langle-1,2,-3\rangle \) ? A. An equation for the plane is \( x+2 y+3 z=14 \). B. An equation for the plane is \( x-2 y+3 z=22 \). C. An equation for the plane is \( -x+2 y-3 z=-10 \). D. An equation for the plane is \( -3 x-2 y+3 z=-10 \).

To find the equation of the plane that passes through the point \( P_{0}(-3,-2,3) \) with the normal vector \( n = \langle -1, 2, -3 \rangle \), we can use the point-normal form of the plane equation: \[ -n_{1}(x - x_0) - n_{2}(y - y_0) - n_{3}(z - z_0) = 0 \] Here, \( n_1 = -1, n_2 = 2, n_3 = -3 \) and \( (x_0, y_0, z_0) = (-3, -2, 3) \). Substituting the values, we have: \[ -(-1)(x + 3) + 2(y + 2) - 3(z - 3) = 0 \] Simplifying this, we obtain: \[ x + 3 + 2y + 4 - 3z + 9 = 0 \] \[ x + 2y - 3z + 16 = 0 \] \[ x + 2y - 3z = -16 \] Now we can compare this with the given options. First, let's get the equations in standard form — they can be manipulated, and it's helpful to express them all in the same format. After checking each option, it becomes evident that the equation rearranging gives you the relationships that match the normal vector \( n \). So now, simplifying the possible answer "C" properly will yield \( -x + 2y - 3z = -10 \), and adding the correct factors and terms provides \( -x + 2y - 3z = -16 \). Hence, the correct answer will fall under option C, tested with additional math techniques or substituting back can confirm further. Happy math exploring! Planes and vectors might seem serious, but there’s always a hidden geometry wonder waiting to be solved! Did you know that understanding planes can significantly help in fields like computer graphics, aerodynamics, and even architecture? The way planes and lines interact is fundamental to designs and structures we use every day! Moreover, a great problem-solving tip is to always double-check your normal vector alignment and ensure that you are consistent with how you set your equations as they can subtly alter your results. Always keep an equation "normalized" if you feel like there's a misalignment!