If `4hati+ 7hatj+ 8hatk, 2hati+ 3hatj+ 4hatk and 2hati+ 5hatj+7hatk` are the position vectors of the vertices A, B and C, respectively, of triangle ABC, then the position vector of the point where the bisector of angle A meets BC is

Options

A. `(2)/(3) (-6hati-8hatj-6hatk)`
B. `(2)/(3)(6hati+8hatj+6hatk)`
C. `(1)/(3) (6hati+ 13hatj+18hatk)`
D. `(1)/(3)(5hatj+12 hatk)`

Explanation

To find the position vector of the point where the angle bisector of angle A meets side BC of triangle ABC, we will follow these steps:

### Step 1: Identify the position vectors of points A, B, and C
Given:
- Position vector of A: \(\vec{A} = 4\hat{i} + 7\hat{j} + 8\hat{k}\)
- Position vector of B: \(\vec{B} = 2\hat{i} + 3\hat{j} + 4\hat{k}\)
- Position vector of C: \(\vec{C} = 2\hat{i} + 5\hat{j} + 7\hat{k}\)

### Step 2: Calculate the vectors AB and AC
The vector \( \vec{AB} \) is given by:
\[ \vec{AB} = \vec{B} - \vec{A} = (2\hat{i} + 3\hat{j} + 4\hat{k}) - (4\hat{i} + 7\hat{j} + 8\hat{k}) = (-2\hat{i} - 4\hat{j} - 4\hat{k}) \]

The vector \( \vec{AC} \) is given by:
\[ \vec{AC} = \vec{C} - \vec{A} = (2\hat{i} + 5\hat{j} + 7\hat{k}) - (4\hat{i} + 7\hat{j} + 8\hat{k}) = (-2\hat{i} - 2\hat{j} - 1\hat{k}) \]

### Step 3: Calculate the magnitudes of vectors AB and AC
The magnitude of \( \vec{AB} \) is:
\[ |\vec{AB}| = \sqrt{(-2)^2 + (-4)^2 + (-4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \]

The magnitude of \( \vec{AC} \) is:
\[ |\vec{AC}| = \sqrt{(-2)^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3 \]

### Step 4: Use the angle bisector theorem to find the position vector of point D
According to the angle bisector theorem, the point D divides BC in the ratio of the lengths of AB and AC:
\[ \text{Ratio} = |\vec{AB}| : |\vec{AC}| = 6 : 3 = 2 : 1 \]

The position vector of point D can be calculated using the formula:
\[ \vec{D} = \frac{|\vec{AC}| \cdot \vec{B} + |\vec{AB}| \cdot \vec{C}}{|\vec{AB}| + |\vec{AC}|} \]

Substituting the values:
\[ \vec{D} = \frac{3 \cdot (2\hat{i} + 3\hat{j} + 4\hat{k}) + 6 \cdot (2\hat{i} + 5\hat{j} + 7\hat{k})}{6 + 3} \]

Calculating the numerator:
\[ = \frac{(6\hat{i} + 9\hat{j} + 12\hat{k}) + (12\hat{i} + 30\hat{j} + 42\hat{k})}{9} = \frac{(18\hat{i} + 39\hat{j} + 54\hat{k})}{9} = 2\hat{i} + \frac{39}{9}\hat{j} + 6\hat{k} = 2\hat{i} + \frac{13}{3}\hat{j} + 6\hat{k} \]

### Final Answer
Thus, the position vector of the point where the bisector of angle A meets BC is:
\[ \vec{D} = 2\hat{i} + \frac{13}{3}\hat{j} + 6\hat{k} \]