(a)Statement 1: Vector `vecc = -5hati + 7 hatj + 2hatk ` is along the bisector of angle between `veca = hati + 2hatj + 2hatk and vecb = 8 hati + hatj - 4hatk`.
Statement 2 : `vecc` is equally inclined to `veca and vecb`.

Options

A. (a) Both the statements are true and statement 2 is the correct explanation for statement 1.
B. (b) Both statements are true but statement 2 is not the correct explanation for statement 1.
C. (c) Statement 1 is true and Statement 2 is false
D. (d) Statement 1 is false and Statement 2 is true.

Explanation

To determine whether the vector \( \vec{c} = -5 \hat{i} + 7 \hat{j} + 2 \hat{k} \) is along the bisector of the angle between the vectors \( \vec{a} = \hat{i} + 2 \hat{j} + 2 \hat{k} \) and \( \vec{b} = 8 \hat{i} + \hat{j} - 4 \hat{k} \), we will follow these steps:

### Step 1: Calculate the magnitudes of vectors \( \vec{a} \) and \( \vec{b} \)

The magnitude of vector \( \vec{a} \) is calculated as follows:
\[ |\vec{a}| = \sqrt{(1)^2 + (2)^2 + (2)^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3 \]

The magnitude of vector \( \vec{b} \) is calculated as follows:
\[ |\vec{b}| = \sqrt{(8)^2 + (1)^2 + (-4)^2} = \sqrt{64 + 1 + 16} = \sqrt{81} = 9 \]

### Step 2: Find the unit vectors of \( \vec{a} \) and \( \vec{b} \)

The unit vector of \( \vec{a} \):
\[ \hat{a} = \frac{\vec{a}}{|\vec{a}|} = \frac{\hat{i} + 2\hat{j} + 2\hat{k}}{3} = \frac{1}{3} \hat{i} + \frac{2}{3} \hat{j} + \frac{2}{3} \hat{k} \]

The unit vector of \( \vec{b} \):
\[ \hat{b} = \frac{\vec{b}}{|\vec{b}|} = \frac{8\hat{i} + \hat{j} - 4\hat{k}}{9} = \frac{8}{9} \hat{i} + \frac{1}{9} \hat{j} - \frac{4}{9} \hat{k} \]

### Step 3: Calculate the bisector vector

The bisector vector \( \vec{d} \) of \( \vec{a} \) and \( \vec{b} \) is given by:
\[ \vec{d} = |\vec{b}|\hat{a} + |\vec{a}|\hat{b} \]
Substituting the values:
\[ \vec{d} = 9\left(\frac{1}{3} \hat{i} + \frac{2}{3} \hat{j} + \frac{2}{3} \hat{k}\right) + 3\left(\frac{8}{9} \hat{i} + \frac{1}{9} \hat{j} - \frac{4}{9} \hat{k}\right) \]
Calculating each term:
\[ \vec{d} = 3\hat{i} + 6\hat{j} + 6\hat{k} + \frac{24}{9} \hat{i} + \frac{3}{9} \hat{j} - \frac{12}{9} \hat{k} \]
\[ = 3\hat{i} + 6\hat{j} + 6\hat{k} + \frac{8}{3} \hat{i} + \frac{1}{3} \hat{j} - \frac{4}{3} \hat{k} \]
Combining the components:
\[ \vec{d} = \left(3 + \frac{8}{3}\right)\hat{i} + \left(6 + \frac{1}{3}\right)\hat{j} + \left(6 - \frac{4}{3}\right)\hat{k} \]
Calculating each component:
\[ \vec{d} = \frac{9}{3} + \frac{8}{3} = \frac{17}{3} \hat{i}, \quad \frac{18}{3} + \frac{1}{3} = \frac{19}{3} \hat{j}, \quad \frac{18}{3} - \frac{4}{3} = \frac{14}{3} \hat{k} \]
Thus,
\[ \vec{d} = \frac{17}{3} \hat{i} + \frac{19}{3} \hat{j} + \frac{14}{3} \hat{k} \]

### Step 4: Compare \( \vec{c} \) with \( \vec{d} \)

Now we need to check if \( \vec{c} \) is a scalar multiple of \( \vec{d} \):
Given \( \vec{c} = -5 \hat{i} + 7 \hat{j} + 2 \hat{k} \).

Since \( \vec{c} \) does not match the direction of \( \vec{d} \), we conclude that:

- **Statement 1** is **false**: \( \vec{c} \) is not along the bisector of \( \vec{a} \) and \( \vec{b} \).

### Step 5: Check Statement 2

To check if \( \vec{c} \) is equally inclined to \( \vec{a} \) and \( \vec{b} \), we need to verify the angles between the vectors. This can be done by checking if the dot products satisfy the condition for equal angles.

Calculating the dot products:
\[ \vec{c} \cdot \vec{a} = (-5)(1) + (7)(2) + (2)(2) = -5 + 14 + 4 = 13 \]
\[ \vec{c} \cdot \vec{b} = (-5)(8) + (7)(1) + (2)(-4) = -40 + 7 - 8 = -41 \]

Since the dot products are not equal, \( \vec{c} \) is not equally inclined to \( \vec{a} \) and \( \vec{b} \).

Thus, **Statement 2** is also **false**.

### Conclusion
Both statements are false.