If `lt l_(1), m_(1), n_(1)gt and lt l_(2), m_(2), n_(2) gt` be the direction cosines of two lines `L_(1) and L_(2)` respectively. If the angle between them is `theta`, the `costheta=?`

Answer

To find the cosine of the angle \( \theta \) between two lines \( L_1 \) and \( L_2 \) given their direction cosines \( (l_1, m_1, n_1) \) and \( (l_2, m_2, n_2) \), we can use the formula for the cosine of the angle between two vectors.

### Step-by-Step Solution:

1. **Understand Direction Cosines**:
The direction cosines of a line are the cosines of the angles that the line makes with the coordinate axes. For lines \( L_1 \) and \( L_2 \), we have:
- For line \( L_1 \): \( (l_1, m_1, n_1) \)
- For line \( L_2 \): \( (l_2, m_2, n_2) \)

2. **Dot Product Formula**:
The cosine of the angle \( \theta \) between two vectors (or lines) can be expressed using the dot product:
\[ \cos \theta = \frac{A \cdot B}{|A| |B|} \]
where \( A \) and \( B \) are the direction vectors of the lines.

3. **Calculate the Dot Product**:
The dot product of the two direction cosines is given by:
\[ A \cdot B = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

4. **Magnitude of Direction Cosines**:
The magnitude of each direction cosine vector is:
\[ |A| = \sqrt{l_1^2 + m_1^2 + n_1^2} \]
\[ |B| = \sqrt{l_2^2 + m_2^2 + n_2^2} \]

5. **Using the Property of Direction Cosines**:
Since \( l_1, m_1, n_1 \) and \( l_2, m_2, n_2 \) are direction cosines, we know:
\[ l_1^2 + m_1^2 + n_1^2 = 1 \]
\[ l_2^2 + m_2^2 + n_2^2 = 1 \]
Therefore, \( |A| = 1 \) and \( |B| = 1 \).

6. **Substituting Values into the Cosine Formula**:
Now substituting these values into the cosine formula:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{1 \cdot 1} \]
Thus, we have:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

### Final Result:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]