If the position vectors of the vertices A, B and C of a $$\triangle$$ABC are respectively $$4\hat{i} + 7\hat{j} + 8\hat{k}$$, $$2\hat{i} + 3\hat{j} + 4\hat{k}$$ and $$2\hat{i} + 5\hat{j} + 7\hat{k}$$, then the position vector of the point, where the bisector of $$\angle A$$ meets BC is:

We have the position vectors of the three vertices written as $$\vec A = 4\hat i + 7\hat j + 8\hat k,\; \vec B = 2\hat i + 3\hat j + 4\hat k,\; \vec C = 2\hat i + 5\hat j + 7\hat k.$$

To locate the point where the internal bisector of $$\angle A$$ meets the side BC, we use the well-known Angle-Bisector Theorem. It states:

$$\frac{BD}{DC} = \frac{AB}{AC},$$

where D is the required point on BC, BD is the length from B to D and DC is the length from D to C.

So we must first compute the lengths $$AB$$ and $$AC.$$

The vector $$\overrightarrow{AB}$$ is obtained by subtracting $$\vec A$$ from $$\vec B$$:

$$\overrightarrow{AB} = \vec B - \vec A = (2-4)\hat i + (3-7)\hat j + (4-8)\hat k = -2\hat i -4\hat j -4\hat k.$$

Now its magnitude is

$$|AB| = \sqrt{(-2)^2 + (-4)^2 + (-4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6.$$

Next, the vector $$\overrightarrow{AC}$$ is

$$\overrightarrow{AC} = \vec C - \vec A = (2-4)\hat i + (5-7)\hat j + (7-8)\hat k = -2\hat i -2\hat j -1\hat k.$$

Its magnitude is

$$|AC| = \sqrt{(-2)^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3.$$

So we obtain

$$\frac{AB}{AC} = \frac{6}{3} = 2,$$

which gives the ratio

$$BD : DC = 2 : 1.$$

Having the ratio, we now apply the Section Formula. If a point D divides the segment joining vectors $$\vec B$$ and $$\vec C$$ internally in the ratio $$m:n$$ (here $$m=2, n=1$$ so that $$BD:DC = 2:1$$), then

$$\vec D = \frac{n\,\vec B + m\,\vec C}{m+n}.$$

Substituting $$m = 2$$ and $$n = 1$$ we get

$$\vec D = \frac{1\cdot\vec B + 2\cdot\vec C}{2+1} = \frac{\vec B + 2\vec C}{3}.$$

First compute $$\vec B + 2\vec C$$:

$$\vec B + 2\vec C = (2\hat i + 3\hat j + 4\hat k) + 2(2\hat i + 5\hat j + 7\hat k)$$

$$= 2\hat i + 3\hat j + 4\hat k + 4\hat i + 10\hat j + 14\hat k$$

$$= (2+4)\hat i + (3+10)\hat j + (4+14)\hat k = 6\hat i + 13\hat j + 18\hat k.$$

Now divide by 3:

$$\vec D = \frac{1}{3}(6\hat i + 13\hat j + 18\hat k).$$

This expression exactly matches the second option provided.

Hence, the correct answer is Option B.