Find the angle between the following pair of lines :
`vec(r) = hati + hatj - hatk + lambda (hati - 3 hatj + 2 hatk)` and
`vec(r) = 2 hat(i) - hat(j) + mu (3 hat(i) + hatj - 2 hatk)`.

Options

A. `pi+cos^(-1) ((2)/(7))`
B. `cos^(-1) ((3)/(7))`
C. ` cos^(-1) ((2)/(7))`
D. `pi - cos^(-1) ((2)/(7))`

Answer

To find the angle between the two given lines represented in vector form, we can follow these steps:

### Step 1: Identify the direction vectors of the lines
The equations of the lines are given as:
1. \(\vec{r_1} = \hat{i} + \hat{j} - \hat{k} + \lambda (\hat{i} - 3\hat{j} + 2\hat{k})\)
2. \(\vec{r_2} = 2\hat{i} - \hat{j} + \mu (3\hat{i} + \hat{j} - 2\hat{k})\)

From these equations, we can extract the direction vectors:
- For the first line, the direction vector \(\vec{b_1} = \hat{i} - 3\hat{j} + 2\hat{k}\)
- For the second line, the direction vector \(\vec{b_2} = 3\hat{i} + \hat{j} - 2\hat{k}\)

### Step 2: Calculate the dot product of the direction vectors
The dot product \(\vec{b_1} \cdot \vec{b_2}\) is calculated as follows:
\[ \vec{b_1} \cdot \vec{b_2} = (1)(3) + (-3)(1) + (2)(-2) \]
Calculating this gives:
\[ = 3 - 3 - 4 = -4 \]

### Step 3: Calculate the magnitudes of the direction vectors
Next, we calculate the magnitudes of \(\vec{b_1}\) and \(\vec{b_2}\):
\[ |\vec{b_1}| = \sqrt{(1)^2 + (-3)^2 + (2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]
\[ |\vec{b_2}| = \sqrt{(3)^2 + (1)^2 + (-2)^2} = \sqrt{9 + 1 + 4} = \sqrt{14} \]

### Step 4: Use the dot product to find the cosine of the angle
Using the dot product and magnitudes, we can find \(\cos \theta\):
\[ \cos \theta = \frac{\vec{b_1} \cdot \vec{b_2}}{|\vec{b_1}| |\vec{b_2}|} \]
Substituting the values:
\[ \cos \theta = \frac{-4}{\sqrt{14} \cdot \sqrt{14}} = \frac{-4}{14} = -\frac{2}{7} \]

### Step 5: Calculate the angle \(\theta\)
To find the angle \(\theta\), we take the inverse cosine:
\[ \theta = \cos^{-1}\left(-\frac{2}{7}\right) \]

### Step 6: Final expression for the angle
Using the property of inverse cosine:
\[ \theta = \pi - \cos^{-1}\left(\frac{2}{7}\right) \]

This gives us the angle between the two lines.