Find the angle between the lines whose direction ratios are:
`2,-3,4 and 1,2,1`.

Explanation

To find the angle between the lines whose direction ratios are given, we can follow these steps:

### Step 1: Identify the direction ratios
The direction ratios for the two lines are:
- Line L1: \( (2, -3, 4) \)
- Line L2: \( (1, 2, 1) \)

### Step 2: Calculate the magnitudes of the direction ratios
The magnitude of a vector \( (a, b, c) \) is given by the formula:
\[ \text{Magnitude} = \sqrt{a^2 + b^2 + c^2} \]

For Line L1:
\[ \text{Magnitude of L1} = \sqrt{2^2 + (-3)^2 + 4^2} = \sqrt{4 + 9 + 16} = \sqrt{29} \]

For Line L2:
\[ \text{Magnitude of L2} = \sqrt{1^2 + 2^2 + 1^2} = \sqrt{1 + 4 + 1} = \sqrt{6} \]

### Step 3: Calculate the direction cosines
The direction cosines are obtained by dividing each direction ratio by the magnitude of the vector.

For Line L1:
\[ l_1 = \frac{2}{\sqrt{29}}, \quad m_1 = \frac{-3}{\sqrt{29}}, \quad n_1 = \frac{4}{\sqrt{29}} \]

For Line L2:
\[ l_2 = \frac{1}{\sqrt{6}}, \quad m_2 = \frac{2}{\sqrt{6}}, \quad n_2 = \frac{1}{\sqrt{6}} \]

### Step 4: Use the formula to find the cosine of the angle
The formula for the cosine of the angle \( \theta \) between two lines is given by:
\[ \cos \theta = l_1 l_2 + m_1 m_2 + n_1 n_2 \]

Substituting the values:
\[ \cos \theta = \left(\frac{2}{\sqrt{29}} \cdot \frac{1}{\sqrt{6}}\right) + \left(\frac{-3}{\sqrt{29}} \cdot \frac{2}{\sqrt{6}}\right) + \left(\frac{4}{\sqrt{29}} \cdot \frac{1}{\sqrt{6}}\right) \]

Calculating each term:
1. \( \frac{2}{\sqrt{29} \sqrt{6}} \)
2. \( \frac{-6}{\sqrt{29} \sqrt{6}} \)
3. \( \frac{4}{\sqrt{29} \sqrt{6}} \)

Now, combine these:
\[ \cos \theta = \frac{2 - 6 + 4}{\sqrt{29} \sqrt{6}} = \frac{0}{\sqrt{29} \sqrt{6}} = 0 \]

### Step 5: Determine the angle
Since \( \cos \theta = 0 \), we have:
\[ \theta = 90^\circ \]

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

---