The angle between the lines `(x)/(1)=(y)/(0)=(z)/(-1)and(x)/(3)=(y)/(4)=(z)/(5)` is

Options

A. `cos^(-1)((1)/(5))`
B. `cos^(-1)((1)/(3))`
C. `cos^(-1)((1)/(2))`
D. `cos^(-1)((1)/(4))`

Answer

To find the angle between the two lines given in the question, we need to first express each line in vector form and then use the formula for the angle between two lines.

### Step 1: Identify the direction ratios of the lines.

The first line is given by:
\[ \frac{x}{1} = \frac{y}{0} = \frac{z}{-1} \]
From this, we can identify the direction ratios as:
- \( l_1 = 1 \)
- \( m_1 = 0 \)
- \( n_1 = -1 \)

Thus, the direction vector of the first line \( \mathbf{A} \) is:
\[ \mathbf{A} = (1, 0, -1) \]

The second line is given by:
\[ \frac{x}{3} = \frac{y}{4} = \frac{z}{5} \]
From this, we can identify the direction ratios as:
- \( l_2 = 3 \)
- \( m_2 = 4 \)
- \( n_2 = 5 \)

Thus, the direction vector of the second line \( \mathbf{B} \) is:
\[ \mathbf{B} = (3, 4, 5) \]

### Step 2: Use the formula for the angle between two lines.

The angle \( \theta \) between two lines can be calculated using the dot product of their direction vectors:
\[ \cos \theta = \frac{\mathbf{A} \cdot \mathbf{B}}{|\mathbf{A}| |\mathbf{B}|} \]

### Step 3: Calculate the dot product \( \mathbf{A} \cdot \mathbf{B} \).

Calculating the dot product:
\[ \mathbf{A} \cdot \mathbf{B} = (1)(3) + (0)(4) + (-1)(5) = 3 + 0 - 5 = -2 \]

### Step 4: Calculate the magnitudes of \( \mathbf{A} \) and \( \mathbf{B} \).

Calculating the magnitude of \( \mathbf{A} \):
\[ |\mathbf{A}| = \sqrt{1^2 + 0^2 + (-1)^2} = \sqrt{1 + 0 + 1} = \sqrt{2} \]

Calculating the magnitude of \( \mathbf{B} \):
\[ |\mathbf{B}| = \sqrt{3^2 + 4^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} = 5\sqrt{2} \]

### Step 5: Substitute into the cosine formula.

Now substituting into the cosine formula:
\[ \cos \theta = \frac{-2}{\sqrt{2} \cdot 5\sqrt{2}} = \frac{-2}{10} = -\frac{1}{5} \]

### Step 6: Calculate the angle \( \theta \).

To find \( \theta \), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(-\frac{1}{5}\right) \]

### Final Answer:
Thus, the angle between the two lines is:
\[ \theta = \cos^{-1}\left(-\frac{1}{5}\right) \]