State Whether TRUE or FALSE:
Angle between the planes :
`vec(r). (hati - 2 hatj - 2 hatk) = 1 and vec(r).(3 hati - 6 hatj + 2 hatk) = 0 ` is `cos^(-1) ((11)/(21))`.

Answer

To determine whether the statement is TRUE or FALSE, we need to find the angle between the two given planes using their normal vectors. Let's break down the solution step by step.

### Step 1: Identify the normal vectors of the planes
The equations of the planes are given as:
1. Plane 1: \(\vec{r} \cdot (\hat{i} - 2\hat{j} - 2\hat{k}) = 1\)
2. Plane 2: \(\vec{r} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 0\)

The normal vector for Plane 1, \( \vec{n_1} \), is \( \hat{i} - 2\hat{j} - 2\hat{k} \) which corresponds to the direction ratios \( (1, -2, -2) \).

The normal vector for Plane 2, \( \vec{n_2} \), is \( 3\hat{i} - 6\hat{j} + 2\hat{k} \) which corresponds to the direction ratios \( (3, -6, 2) \).

### Step 2: Calculate the dot product of the normal vectors
The dot product \( \vec{n_1} \cdot \vec{n_2} \) is calculated as follows:

\[ \vec{n_1} \cdot \vec{n_2} = (1)(3) + (-2)(-6) + (-2)(2) \]
\[ = 3 + 12 - 4 = 11 \]

### Step 3: Calculate the magnitudes of the normal vectors
Now, we need to find the magnitudes of \( \vec{n_1} \) and \( \vec{n_2} \).

Magnitude of \( \vec{n_1} \):
\[ |\vec{n_1}| = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

Magnitude of \( \vec{n_2} \):
\[ |\vec{n_2}| = \sqrt{3^2 + (-6)^2 + 2^2} = \sqrt{9 + 36 + 4} = \sqrt{49} = 7 \]

### Step 4: Calculate the cosine of the angle between the planes
Using the formula for the cosine of the angle \( \theta \) between the two planes:
\[ \cos \theta = \frac{\vec{n_1} \cdot \vec{n_2}}{|\vec{n_1}| |\vec{n_2}|} \]
Substituting the values we calculated:
\[ \cos \theta = \frac{11}{3 \times 7} = \frac{11}{21} \]

### Step 5: Determine the angle
Thus, we have:
\[ \theta = \cos^{-1}\left(\frac{11}{21}\right) \]

### Conclusion
The statement claims that the angle between the planes is \( \cos^{-1}\left(\frac{11}{21}\right) \). Since we have calculated and confirmed this value, the statement is TRUE.