Find the direction cosines of the line perpendicular to the lines whose direction ratios are `-2, 1, -1 and -3, -4,1`.

Explanation

To find the direction cosines of the line that is perpendicular to the lines with direction ratios \((-2, 1, -1)\) and \((-3, -4, 1)\), we can follow these steps:

### Step 1: Identify the Direction Ratios
Let the direction ratios of the first line be \( \mathbf{A} = (-2, 1, -1) \) and the second line be \( \mathbf{B} = (-3, -4, 1) \).

### Step 2: Set Up the Perpendicularity Condition
For two lines to be perpendicular, the dot product of their direction ratios must equal zero. If \( \mathbf{P} = (a, b, c) \) represents the direction ratios of the line we want to find, we can set up the following equations based on the dot product:

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