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 use the formula for the cosine of the angle \( \theta \) between two lines:

\[ \cos \theta = |L_1 L_2 + M_1 M_2 + N_1 N_2| \]

where \( (L_1, M_1, N_1) \) and \( (L_2, M_2, N_2) \) are the direction cosines of the two lines.

### Step 1: Identify the direction cosines
From the problem, we have:
- For the first line:
\[ L_1 = -\frac{\sqrt{3}}{4}, \quad M_1 = \frac{1}{4}, \quad N_1 = -\frac{\sqrt{3}}{2} \]
- For the second line:
\[ L_2 = -\frac{\sqrt{3}}{4}, \quad M_2 = \frac{1}{4}, \quad N_2 = \frac{\sqrt{3}}{2} \]

### Step 2: Substitute the values into the formula
Now we substitute these values into the cosine formula:
\[ \cos \theta = \left| L_1 L_2 + M_1 M_2 + N_1 N_2 \right| \]
Calculating each term:
- \( L_1 L_2 = \left(-\frac{\sqrt{3}}{4}\right) \left(-\frac{\sqrt{3}}{4}\right) = \frac{3}{16} \)
- \( M_1 M_2 = \left(\frac{1}{4}\right) \left(\frac{1}{4}\right) = \frac{1}{16} \)
- \( N_1 N_2 = \left(-\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) = -\frac{3}{4} \)

### Step 3: Combine the results
Now, we combine these results:
\[ \cos \theta = \left| \frac{3}{16} + \frac{1}{16} - \frac{3}{4} \right| \]
We need a common denominator to combine these fractions. The common denominator is 16:
\[ -\frac{3}{4} = -\frac{12}{16} \]
Thus,
\[ \cos \theta = \left| \frac{3}{16} + \frac{1}{16} - \frac{12}{16} \right| = \left| \frac{4}{16} - \frac{12}{16} \right| = \left| -\frac{8}{16} \right| = \frac{8}{16} = \frac{1}{2} \]

### Step 4: Find the angle \( \theta \)
Now, we know that:
\[ \cos \theta = \frac{1}{2} \]
The angle \( \theta \) for which \( \cos \theta = \frac{1}{2} \) is:
\[ \theta = \frac{\pi}{3} \text{ radians} \]

### Final Answer
The angle between the lines is \( \frac{\pi}{3} \) radians.

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