Find the angle between the pairs of line `r=3hat(i)+2hat(j)-4hat(k)+lambda(hat(i)+2hat(j)+2hat(k)) and hat(r)=5hat(i)-2hat(j)+mu(3hat(i)+2hat(j)+6hat(k))`.

Answer

To find the angle between the given lines, we will follow these steps:

### Step 1: Identify the direction ratios of the lines
The equations of the lines are given as:
1. \( \mathbf{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k}) \)
2. \( \mathbf{r} = 5\hat{i} - 2\hat{j} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k}) \)

From these equations, we can extract the direction ratios:
- For the first line, the direction ratios are \( \mathbf{a} = \hat{i} + 2\hat{j} + 2\hat{k} \) which gives us \( (1, 2, 2) \).
- For the second line, the direction ratios are \( \mathbf{b} = 3\hat{i} + 2\hat{j} + 6\hat{k} \) which gives us \( (3, 2, 6) \).

### Step 2: Calculate the dot product of the direction ratios
The dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as follows:
\[ \mathbf{a} \cdot \mathbf{b} = (1)(3) + (2)(2) + (2)(6) = 3 + 4 + 12 = 19 \]

### Step 3: Calculate the magnitudes of the direction ratios
Now we calculate the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \):
- Magnitude of \( \mathbf{a} \):
\[ |\mathbf{a}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]
- Magnitude of \( \mathbf{b} \):
\[ |\mathbf{b}| = \sqrt{3^2 + 2^2 + 6^2} = \sqrt{9 + 4 + 36} = \sqrt{49} = 7 \]

### Step 4: Use the dot product to find the cosine of the angle
Using the formula for the cosine of the angle \( \theta \) between two vectors:
\[ \cos \theta = \frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}| |\mathbf{b}|} \]
Substituting the values we calculated:
\[ \cos \theta = \frac{19}{3 \times 7} = \frac{19}{21} \]

### Step 5: Calculate the angle \( \theta \)
Finally, we find the angle \( \theta \) by taking the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{19}{21}\right) \]

This gives us the angle between the two lines.

### Summary of the Solution:
The angle between the given lines is \( \theta = \cos^{-1}\left(\frac{19}{21}\right) \).

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