STATEMENT-1 : The vector `hati` bisects the angle between the vectors `hati-2hatj-2hatk` and `hati+2hatj+2hatk`.
And
STATEMENT-2 : The vector along the angle bisector of the vector `veca` and `vecb` is given by `+-((veca)/(|veca|)+-(vecb)/(|vecb|))` where `|veca|.|vecb|ne 0`

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-3
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-3
C. Statement-1 is True, Statement-2 is False
D. Statement-1 is False, Statement-2 is True

Answer

To solve the problem, we will analyze both statements step by step.

### Step 1: Analyze Statement 1
We need to check if the vector \(\hat{i}\) bisects the angle between the vectors \(\vec{a} = \hat{i} - 2\hat{j} - 2\hat{k}\) and \(\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}\).

1. **Find the direction of the vectors**:
- \(\vec{a} = \hat{i} - 2\hat{j} - 2\hat{k}\)
- \(\vec{b} = \hat{i} + 2\hat{j} + 2\hat{k}\)

2. **Calculate the magnitudes**:
- \(|\vec{a}| = \sqrt{1^2 + (-2)^2 + (-2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\)
- \(|\vec{b}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3\)

3. **Normalize the vectors**:
- \(\hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{\hat{i} - 2\hat{j} - 2\hat{k}}{3} = \frac{1}{3}\hat{i} - \frac{2}{3}\hat{j} - \frac{2}{3}\hat{k}\)
- \(\hat{b} = \frac{\vec{b}}{|\vec{b}|} = \frac{\hat{i} + 2\hat{j} + 2\hat{k}}{3} = \frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\)

4. **Find the angle bisector**:
The angle bisector \(\hat{t}\) can be given by:
\[ \hat{t} = \hat{a} + \hat{b} = \left(\frac{1}{3}\hat{i} - \frac{2}{3}\hat{j} - \frac{2}{3}\hat{k}\right) + \left(\frac{1}{3}\hat{i} + \frac{2}{3}\hat{j} + \frac{2}{3}\hat{k}\right) \]
Simplifying this:
\[ \hat{t} = \frac{2}{3}\hat{i} + 0\hat{j} + 0\hat{k} = \frac{2}{3}\hat{i} \]

5. **Check if \(\hat{i}\) is a scalar multiple of \(\hat{t}\)**:
The vector \(\hat{i}\) is indeed a scalar multiple of \(\hat{t}\) since \(\hat{t} = \frac{2}{3}\hat{i}\). Thus, \(\hat{i}\) bisects the angle between \(\vec{a}\) and \(\vec{b}\).

### Conclusion for Statement 1:
**Statement 1 is True.**

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### Step 2: Analyze Statement 2
The statement claims that the vector along the angle bisector of vectors \(\vec{a}\) and \(\vec{b}\) is given by:
\[ \pm\left(\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}\right) \]

1. **Substituting the normalized vectors**:
From our previous calculations, we have:
\[ \hat{a} + \hat{b} = \frac{2}{3}\hat{i} \]

2. **Check the given formula**:
The formula states:
\[ \hat{t} = \pm\left(\frac{\hat{i} - 2\hat{j} - 2\hat{k}}{3} + \frac{\hat{i} + 2\hat{j} + 2\hat{k}}{3}\right) \]
Simplifying this gives:
\[ \hat{t} = \pm\left(\frac{2}{3}\hat{i}\right) \]
This matches our previous result, but the statement also implies that it holds for all cases where \(|\vec{a}| \cdot |\vec{b}| \neq 0\).

### Conclusion for Statement 2:
**Statement 2 is True.**

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### Final Conclusion:
- **Statement 1 is True.**
- **Statement 2 is True.**