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)`
The angle between the lines whose direction
cosines are `(1/2, 1/2,1/sqrt(2)) and (-1/2, -1/2, 1/sqrt(2))` is

Options

A. `0^(@)`
B. `60^(@)`
C. `90^(@)`
D. `120^(@)`

Answer

To find the angle between the two lines whose direction cosines are given, we can use the formula for the cosine of the angle between two lines based on their direction cosines.

Given:
- Direction cosines of the first line: \( (l_1, m_1, n_1) = \left(\frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}}\right) \)
- Direction cosines of the second line: \( (l_2, m_2, n_2) = \left(-\frac{1}{2}, -\frac{1}{2}, \frac{1}{\sqrt{2}}\right) \)

### Step 1: Calculate \( \cos \theta \)
Using the formula:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

Substituting the values:
\[ \cos \theta = \left(\frac{1}{2} \cdot -\frac{1}{2}\right) + \left(\frac{1}{2} \cdot -\frac{1}{2}\right) + \left(\frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}}\right) \]

### Step 2: Simplify the expression
Calculating each term:
- First term: \( \frac{1}{2} \cdot -\frac{1}{2} = -\frac{1}{4} \)
- Second term: \( \frac{1}{2} \cdot -\frac{1}{2} = -\frac{1}{4} \)
- Third term: \( \frac{1}{\sqrt{2}} \cdot \frac{1}{\sqrt{2}} = \frac{1}{2} \)

Now substitute these back into the equation:
\[ \cos \theta = -\frac{1}{4} - \frac{1}{4} + \frac{1}{2} \]

### Step 3: Combine the terms
Combining the terms:
\[ \cos \theta = -\frac{1}{4} - \frac{1}{4} + \frac{2}{4} = -\frac{1}{2} + \frac{2}{4} = 0 \]

### Step 4: Determine the angle \( \theta \)
Since \( \cos \theta = 0 \), we find:
\[ \theta = \cos^{-1}(0) = 90^\circ \]

### Conclusion
The angle between the two lines is \( 90^\circ \), indicating that the lines are perpendicular to each other.

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