If `l_(1), m_(1), n_(1)` and `l_(2),m_(2),n_(2)` are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are `m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1)`.

Explanation

To show that the direction cosines of the line perpendicular to two mutually perpendicular lines with direction cosines \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are given by \( m_1 n_2 - m_2 n_1, n_1 l_2 - n_2 l_1, l_1 m_2 - l_2 m_1 \), we can follow these steps:

### Step 1: Define the Direction Cosines
Let the direction cosines of the first line be \( l_1, m_1, n_1 \) and for the second line be \( l_2, m_2, n_2 \).

### Step 2: Represent the Lines as Vectors
We can represent these lines as vectors:
- For the first line:
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