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, we can use the point-normal form of a plane's equation, which is given by: \[ n_x(x - x_0) + n_y(y - y_0) + n_z(z - z_0) = 0 \] In this case, the normal vector \(n\) is \(\langle -1, 2, -3 \rangle\) and the point \(P_0(-3, -2, 3)\) gives us \(x_0 = -3\), \(y_0 = -2\), and \(z_0 = 3\). Substituting these values into the equation: \[ -1(x - (-3)) + 2(y - (-2)) - 3(z - 3) = 0 \] \[ -1(x + 3) + 2(y + 2) - 3(z - 3) = 0 \] \[ -x - 3 + 2y + 4 - 3z + 9 = 0 \] \[ -x + 2y - 3z + 10 = 0 \] Rearranging gives: \[ -x + 2y - 3z = -10 \] So the equation of the plane is \( -x + 2y - 3z = -10 \). Now, let's check the given options: A. \( x + 2y + 3z = 14 \) (Not correct) B. \( x - 2y + 3z = 22 \) (Not correct) C. \( -x + 2y - 3z = -10 \) (Correct) D. \( -3x - 2y + 3z = -10 \) (Not correct) The correct answer is C: An equation for the plane is \( -x + 2y - 3z = -10 \). Now, here you go with some extra goodies to keep things interesting! Every plane can be thought of as having infinite friends! You can create a family of parallel planes by simply tweaking the constant term of the equation. This allows planes to have the same tilt and angle but different positions in space. Imagine them as parallel layers of cake, each deliciously stacked, while still maintaining that perfect balance of ingredients! When solving for the equation of a plane, it’s easy to overlook the signs and flipping the normal vector. Remember, both \( n \) and \(-n\) can define the same plane, but with different formulations. A common mistake is forgetting that the normal vector matters! Miss one sign, and suddenly, you might be grappling with an altogether different plane. Keeping your signs straight is essential, like balancing on a tightrope between the realms of geometry and algebra!