What is the angle between the lines whose direction cosines are proportional to (2,3,4) and (1,-2,1) respectively?

Options

A. `90^@`
B. `60^@`
C. `45^@`
D. `30^@`

Explanation

To find the angle between two lines whose direction cosines are proportional to (2, 3, 4) and (1, -2, 1), we can use the formula for the cosine of the angle between two vectors.

### Step-by-Step Solution:

1. **Identify the Direction Cosines**:
Let the direction cosines of the first line be \( l_1 = 2 \), \( m_1 = 3 \), \( n_1 = 4 \) and for the second line \( l_2 = 1 \), \( m_2 = -2 \), \( n_2 = 1 \).

2. **Use the Dot Product Formula**:
The angle \( \theta \) between two lines can be found using the dot product formula:
\[ \cos \theta = \frac{l_1 l_2 + m_1 m_2 + n_1 n_2}{\sqrt{l_1^2 + m_1^2 + n_1^2} \sqrt{l_2^2 + m_2^2 + n_2^2}} \]

3. **Calculate the Dot Product**:
\[ l_1 l_2 + m_1 m_2 + n_1 n_2 = (2)(1) + (3)(-2) + (4)(1) = 2 - 6 + 4 = 0 \]

4. **Calculate the Magnitudes**:
- For the first line:
\[ \sqrt{l_1^2 + m_1^2 + n_1^2} = \sqrt{2^2 + 3^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \]
- For the second line:
\[ \sqrt{l_2^2 + m_2^2 + n_2^2} = \sqrt{1^2 + (-2)^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

5. **Substitute into the Cosine Formula**:
\[ \cos \theta = \frac{0}{\sqrt{29} \cdot \sqrt{6}} = 0 \]

6. **Determine the Angle**:
Since \( \cos \theta = 0 \), this means:
\[ \theta = 90^\circ \]

### Final Answer:
The angle between the lines is \( 90^\circ \).