If direction ratios of two lines are `5,-12,13 and -3,4,5`, then the angle between them is

Options

A. `cos^(-1)((1)/(65))`
B. `cos^(-1)((2)/(65))`
C. `cos^(-1)((3)/(65))`
D. `(pi)/(2)`

Answer

To find the angle between two lines given their direction ratios, we can use the formula for the cosine of the angle \( \theta \) between two vectors. The direction ratios of the first line are \( (A_1, B_1, C_1) = (5, -12, 13) \) and for the second line, they are \( (A_2, B_2, C_2) = (-3, 4, 5) \).

### Step-by-Step Solution:

1. **Calculate the dot product of the direction ratios**:
\[ A_1 A_2 + B_1 B_2 + C_1 C_2 = 5 \cdot (-3) + (-12) \cdot 4 + 13 \cdot 5 \]
\[ = -15 - 48 + 65 = 2 \]

2. **Calculate the magnitudes of the direction ratios**:
\[ |A_1, B_1, C_1| = \sqrt{A_1^2 + B_1^2 + C_1^2} = \sqrt{5^2 + (-12)^2 + 13^2} \]
\[ = \sqrt{25 + 144 + 169} = \sqrt{338} \]

\[ |A_2, B_2, C_2| = \sqrt{A_2^2 + B_2^2 + C_2^2} = \sqrt{(-3)^2 + 4^2 + 5^2} \]
\[ = \sqrt{9 + 16 + 25} = \sqrt{50} \]

3. **Substitute the values into the cosine formula**:
\[ \cos \theta = \frac{A_1 A_2 + B_1 B_2 + C_1 C_2}{|A_1, B_1, C_1| \cdot |A_2, B_2, C_2|} \]
\[ = \frac{2}{\sqrt{338} \cdot \sqrt{50}} = \frac{2}{\sqrt{16900}} = \frac{2}{130} \]
\[ = \frac{1}{65} \]

4. **Find the angle \( \theta \)**:
\[ \theta = \cos^{-1}\left(\frac{1}{65}\right) \]

### Final Answer:
The angle between the two lines is \( \theta = \cos^{-1}\left(\frac{1}{65}\right) \).