Find the angle between the lines whose direction cosines are `(-(sqrt3)/(4), (1)/(4), -(sqrt3)/(2)) and (-(sqrt3)/(4), (1)/(4), (sqrt3)/(2))`.

Answer

To find the angle between the lines whose direction cosines are given, we can follow these steps:

### Step 1: Identify the direction cosines
The direction cosines of the first line are:
- \( L_1 = -\frac{\sqrt{3}}{4} \)
- \( M_1 = \frac{1}{4} \)
- \( N_1 = -\frac{\sqrt{3}}{2} \)

The direction cosines of the second line are:
- \( L_2 = -\frac{\sqrt{3}}{4} \)
- \( M_2 = \frac{1}{4} \)
- \( N_2 = \frac{\sqrt{3}}{2} \)

### Step 2: Use the formula for the cosine of the angle between two lines
The formula for the cosine of the angle \( \theta \) between two lines with direction cosines \( (L_1, M_1, N_1) \) and \( (L_2, M_2, N_2) \) is given by:
\[ \cos \theta = L_1 L_2 + M_1 M_2 + N_1 N_2 \]

### Step 3: Substitute the values into the formula
Substituting the values of the direction cosines into the formula:
\[ \cos \theta = \left(-\frac{\sqrt{3}}{4}\right)\left(-\frac{\sqrt{3}}{4}\right) + \left(\frac{1}{4}\right)\left(\frac{1}{4}\right) + \left(-\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) \]

### Step 4: Calculate each term
Calculating each term:
1. \( L_1 L_2 = \left(-\frac{\sqrt{3}}{4}\right)\left(-\frac{\sqrt{3}}{4}\right) = \frac{3}{16} \)
2. \( M_1 M_2 = \left(\frac{1}{4}\right)\left(\frac{1}{4}\right) = \frac{1}{16} \)
3. \( N_1 N_2 = \left(-\frac{\sqrt{3}}{2}\right)\left(\frac{\sqrt{3}}{2}\right) = -\frac{3}{4} = -\frac{12}{16} \)

### Step 5: Combine the results
Now, combine the results:
\[ \cos \theta = \frac{3}{16} + \frac{1}{16} - \frac{12}{16} = \frac{3 + 1 - 12}{16} = \frac{-8}{16} = -\frac{1}{2} \]

### Step 6: Find the angle \( \theta \)
To find the angle \( \theta \), we use the cosine value:
\[ \cos \theta = -\frac{1}{2} \]
The angle \( \theta \) corresponding to \( \cos \theta = -\frac{1}{2} \) is:
\[ \theta = 120^\circ \quad \text{(or } \frac{2\pi}{3} \text{ radians)} \]

### Final Answer
Thus, the angle between the lines is:
\[ \theta = 120^\circ \]

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