A horizontal plane `4x-3y+7z=0` is given. Find a line of greatest slope passes through the point `(2,1,1)` in the plane `2x+y-5z=0.`

Explanation

To solve the problem, we need to find a line of greatest slope that passes through the point (2, 1, 1) and lies in the plane defined by the equation \(2x + y - 5z = 0\). The line of greatest slope will be perpendicular to the line of intersection of the two planes: the given horizontal plane \(4x - 3y + 7z = 0\) and the plane \(2x + y - 5z = 0\).

### Step-by-Step Solution:

1. **Identify the Normal Vectors of the Planes**:
- The normal vector of the first plane \(4x - 3y + 7z = 0\) is \(\mathbf{n_1} = (4, -3, 7)\).
- The normal vector of the second plane \(2x + y - 5z = 0\) is \(\mathbf{n_2} = (2, 1, -5)\).

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