Question

Find the cartesian equation of the plane passing through \( A(1, 2, 3) \) and the direction ratios of whose normal are 3, 2, 5.

Hint:

Plane equation uses normal vector and a point: \( a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 \).

Answer 1

The equation of a plane with normal direction ratios \( (a, b, c) \) and passing through \( (x_0, y_0, z_0) \) is: 
\[ a(x - x_0) + b(y - y_0) + c(z - z_0) = 0. \] Given: Point \( (1, 2, 3) \), normal direction ratios \( (3, 2, 5) \). 
\[ 3(x - 1) + 2(y - 2) + 5(z - 3) = 0. \] \[ 3x - 3 + 2y - 4 + 5z - 15 = 0 \Rightarrow 3x + 2y + 5z - 22 = 0. \] Answer: \( 3x + 2y + 5z - 22 = 0 \).