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_1n_2-m_2n_1, n_1l_2-n_2l_1, l_1m_2-l_2m_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 \( m_1n_2 - m_2n_1, n_1l_2 - n_2l_1, l_1m_2 - l_2m_1 \), we can follow these steps:

### Step 1: Define the vectors
Let the direction vectors of the two lines be defined as:
- Line OA: \( \vec{OA} = l_1 \hat{i} + m_1 \hat{j} + n_1 \hat{k} \)
- Line OB: \( \vec{OB} = l_2 \hat{i} + m_2 \hat{j} + n_2 \hat{k} \)

### Step 2: Find the cross product
...