Find the obtuse angle between two lines whose direction-ratios are :
`lt 3, - 6 , 2 gt and lt 1 , -2, -2 gt `.

Answer

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

### Step 1: Identify the Direction Ratios
The direction ratios of the first line are \( \mathbf{L} = \langle 3, -6, 2 \rangle \) and for the second line, they are \( \mathbf{M} = \langle 1, -2, -2 \rangle \).

### Step 2: Calculate the Dot Product
The dot product \( \mathbf{L} \cdot \mathbf{M} \) is calculated as follows:
\[ \mathbf{L} \cdot \mathbf{M} = 3 \cdot 1 + (-6) \cdot (-2) + 2 \cdot (-2) \]
Calculating each term:
- \( 3 \cdot 1 = 3 \)
- \( -6 \cdot -2 = 12 \)
- \( 2 \cdot -2 = -4 \)

Now, summing these results:
\[ \mathbf{L} \cdot \mathbf{M} = 3 + 12 - 4 = 11 \]

### Step 3: Calculate the Magnitudes of the Vectors
Now, we need to find the magnitudes of both vectors \( \mathbf{L} \) and \( \mathbf{M} \).

For \( \mathbf{L} \):
\[ |\mathbf{L}| = \sqrt{3^2 + (-6)^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

For \( \mathbf{M} \):
\[ |\mathbf{M}| = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

### Step 4: Calculate the Cosine of the Angle
Using the formula for the cosine of the angle \( \theta \) between two vectors:
\[ \cos \theta = \frac{\mathbf{L} \cdot \mathbf{M}}{|\mathbf{L}| |\mathbf{M}|} \]
Substituting the values we found:
\[ \cos \theta = \frac{11}{7 \cdot 3} = \frac{11}{21} \]

### Step 5: Calculate the Angle
To find the angle \( \theta \):
\[ \theta = \cos^{-1} \left( \frac{11}{21} \right) \]

### Step 6: Find the Obtuse Angle
Since we are looking for the obtuse angle, we can find it using:
\[ \text{Obtuse angle} = 180^\circ - \theta \]

### Final Answer
Thus, the obtuse angle between the two lines is:
\[ \text{Obtuse angle} = 180^\circ - \cos^{-1} \left( \frac{11}{21} \right) \]