If the direction cosines of two lines are `(l_(1), m_(1), n_(1))` and
`(l_(2), m_(2), n_(2))` and the angle between them is `theta` then
`l_(1)^(2)+m_(1)^(2)+n_(1)^(2)=1=l_(2)^(2)+m_(2)^(2)+n_(2)^(2)`
and costheta` = l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2)`
If the angle between the lines is `60^(@)` then the value
of `l_(1)(l_(1)+l_(2))+m_(1)(m_(1)+m_(2))+n_(1)(n_(1)+n_(2))` is

Options

A. 0
B. `3/2`
C. `1/2`
D. 1

Answer

To solve the problem step by step, we will use the given information about the direction cosines of the lines and the angle between them.

### Step 1: Understand the given information
We have two lines with direction cosines:
- Line 1: \( (l_1, m_1, n_1) \)
- Line 2: \( (l_2, m_2, n_2) \)

It is given that:
1. \( l_1^2 + m_1^2 + n_1^2 = 1 \)
2. \( l_2^2 + m_2^2 + n_2^2 = 1 \)
3. The angle \( \theta \) between the lines is \( 60^\circ \), which means \( \cos \theta = \cos 60^\circ = \frac{1}{2} \).

### Step 2: Use the cosine formula
From the cosine formula for the angle between two lines, we have:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]
Substituting \( \cos 60^\circ \):
\[ \frac{1}{2} = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

### Step 3: Set up the expression to find
We need to find the value of:
\[ l_1(l_1 + l_2) + m_1(m_1 + m_2) + n_1(n_1 + n_2) \]

### Step 4: Expand the expression
Expanding the expression gives:
\[ l_1^2 + l_1 l_2 + m_1^2 + m_1 m_2 + n_1^2 + n_1 n_2 \]

### Step 5: Substitute known values
We know from the given information:
- \( l_1^2 + m_1^2 + n_1^2 = 1 \)

Thus, we can substitute this into our expression:
\[ 1 + l_1 l_2 + m_1 m_2 + n_1 n_2 \]

### Step 6: Substitute the value of \( l_1 l_2 + m_1 m_2 + n_1 n_2 \)
From Step 2, we have:
\[ l_1 l_2 + m_1 m_2 + n_1 n_2 = \frac{1}{2} \]

### Step 7: Final calculation
Now substituting this value back into our expression:
\[ 1 + \frac{1}{2} = \frac{3}{2} \]

### Conclusion
Thus, the value of \( l_1(l_1 + l_2) + m_1(m_1 + m_2) + n_1(n_1 + n_2) \) is \( \frac{3}{2} \).