An angle between the lines whose direction number are 1, -2, 1 and -6, -1, 4 is

Options

A. `(pi)/(6)`
B. `(pi)/(4)`
C. `(pi)/(3)`
D. `(pi)/(2)`

Answer

To find the angle between the two lines with direction numbers (or ratios) given as \( (1, -2, 1) \) and \( (-6, -1, 4) \), we can use the formula for the cosine of the angle \( \theta \) between two lines in three-dimensional space. The formula is:

\[ \cos \theta = \frac{A_1 A_2 + B_1 B_2 + C_1 C_2}{\sqrt{A_1^2 + B_1^2 + C_1^2} \cdot \sqrt{A_2^2 + B_2^2 + C_2^2}} \]

### Step 1: Identify the direction ratios
Let:
- \( A_1 = 1, B_1 = -2, C_1 = 1 \)
- \( A_2 = -6, B_2 = -1, C_2 = 4 \)

### Step 2: Calculate the dot product \( A_1 A_2 + B_1 B_2 + C_1 C_2 \)
\[ A_1 A_2 = 1 \cdot (-6) = -6 \]
\[ B_1 B_2 = -2 \cdot (-1) = 2 \]
\[ C_1 C_2 = 1 \cdot 4 = 4 \]
Now, sum these values:
\[ A_1 A_2 + B_1 B_2 + C_1 C_2 = -6 + 2 + 4 = 0 \]

### Step 3: Calculate the magnitudes of the direction ratios
First, calculate the magnitude of the first line:
\[ \sqrt{A_1^2 + B_1^2 + C_1^2} = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

Now, calculate the magnitude of the second line:
\[ \sqrt{A_2^2 + B_2^2 + C_2^2} = \sqrt{(-6)^2 + (-1)^2 + 4^2} = \sqrt{36 + 1 + 16} = \sqrt{53} \]

### Step 4: Substitute into the cosine formula
Now substitute back into the cosine formula:
\[ \cos \theta = \frac{0}{\sqrt{6} \cdot \sqrt{53}} = 0 \]

### Step 5: Find the angle \( \theta \)
Since \( \cos \theta = 0 \), we find:
\[ \theta = \cos^{-1}(0) = \frac{\pi}{2} \]

### Conclusion
The angle between the two lines is \( \frac{\pi}{2} \).