The angle between the lines `overset(to)( r)=(4 hat(i) - hat(j) )+ lambda(2 hat(i) + hat(j) - 3hat(k) ) and overset(to) (r )=( hat(i) -hat(j) + 2 hat(k) ) + mu (hat(i) - 3 hat(j) - 2 hat(k) )` is

Options

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

Answer

To find the angle between the given lines represented by the vector equations, we will follow these steps:

### Step 1: Identify Direction Vectors
The vector equations of the lines are given as:
1. \(\overset{\to}{r_1} = (4\hat{i} - \hat{j}) + \lambda(2\hat{i} + \hat{j} - 3\hat{k})\)
2. \(\overset{\to}{r_2} = (\hat{i} - \hat{j} + 2\hat{k}) + \mu(\hat{i} - 3\hat{j} - 2\hat{k})\)

From these equations, we can identify the direction vectors:
- For the first line, \(\overset{\to}{L_1} = 2\hat{i} + \hat{j} - 3\hat{k}\)
- For the second line, \(\overset{\to}{L_2} = \hat{i} - 3\hat{j} - 2\hat{k}\)

### Step 2: Calculate the Dot Product of Direction Vectors
The dot product of the vectors \(\overset{\to}{L_1}\) and \(\overset{\to}{L_2}\) is calculated as follows:
\[ \overset{\to}{L_1} \cdot \overset{\to}{L_2} = (2\hat{i} + \hat{j} - 3\hat{k}) \cdot (\hat{i} - 3\hat{j} - 2\hat{k}) \]
Calculating the dot product:
\[ = 2 \cdot 1 + 1 \cdot (-3) + (-3) \cdot (-2) \]
\[ = 2 - 3 + 6 = 5 \]

### Step 3: Calculate the Magnitudes of the Direction Vectors
Now, we need to find the magnitudes of both direction vectors:
1. Magnitude of \(\overset{\to}{L_1}\):
\[ |\overset{\to}{L_1}| = \sqrt{2^2 + 1^2 + (-3)^2} = \sqrt{4 + 1 + 9} = \sqrt{14} \]

2. Magnitude of \(\overset{\to}{L_2}\):
\[ |\overset{\to}{L_2}| = \sqrt{1^2 + (-3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]

### Step 4: Use the Formula for the Angle Between Two Lines
The cosine of the angle \(\theta\) between the two lines can be calculated using the formula:
\[ \cos \theta = \frac{\overset{\to}{L_1} \cdot \overset{\to}{L_2}}{|\overset{\to}{L_1}| \cdot |\overset{\to}{L_2}|} \]
Substituting the values we found:
\[ \cos \theta = \frac{5}{\sqrt{14} \cdot \sqrt{14}} = \frac{5}{14} \]

### Step 5: Find the Angle \(\theta\)
Finally, we find the angle \(\theta\) by taking the inverse cosine:
\[ \theta = \cos^{-1}\left(\frac{5}{14}\right) \]

### Final Answer
The angle between the two lines is \(\theta = \cos^{-1}\left(\frac{5}{14}\right)\).

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