Find the angle between the lines `x-2y+z=0=x+2y-2za n dx+2y+z=0=3x+9y+5zdot`

Explanation

To find the angle between the lines formed by the intersection of two planes given by the equations \(x - 2y + z = 0\) and \(x + 2y - 2z = 0\) and the equations \(dx + 2y + z = 0\) and \(3x + 9y + 5z = 0\), we will follow these steps:

### Step 1: Identify the normal vectors of the planes
The normal vector of a plane given by the equation \(Ax + By + Cz = D\) is given by the vector \((A, B, C)\).

1. For the plane \(x - 2y + z = 0\), the normal vector is \( \mathbf{n_1} = (1, -2, 1) \).
2. For the plane \(x + 2y - 2z = 0\), the normal vector is \( \mathbf{n_2} = (1, 2, -2) \).
3. For the plane \(dx + 2y + z = 0\), the normal vector is \( \mathbf{n_3} = (d, 2, 1) \).
...