1. Key Concepts and Formulas

• Angle Bisector Theorem: In a triangle $\Delta ABC$, if the internal bisector of $\angle A$ intersects side $BC$ at point $D$, then the ratio of the lengths of the segments $BD$ and $DC$ is equal to the ratio of the lengths of the adjacent sides $AB$ and $AC$. Mathematically, $\frac{BD}{DC} = \frac{AB}{AC}$.
• Section Formula (Internal Division): If a point $D$ divides the line segment joining points $B$ and $C$ internally in the ratio $m:n$, then the position vector of $D$, denoted by $\vec{d}$, is given by $\vec{d} = \frac{n\vec{b} + m\vec{c}}{m+n}$, where $\vec{b}$ and $\vec{c}$ are the position vectors of $B$ and $C$, respectively.

1. Step-by-Step Solution

• Step 1: Understand the Problem and Identify Given Information We are given the position vectors of the vertices $A$, $B$, and $C$ of a triangle $\Delta ABC$: $\vec{a} = 4\widehat i + 7\widehat j + 8\widehat k$ $\vec{b} = 2\widehat i + 3\widehat j + 4\widehat k$ $\vec{c} = 2\widehat i + 5\widehat j + 7\widehat k$ We need to find the position vector of the point where the bisector of $\angle A$ meets side $BC$. Let this point be $D$.

• Step 2: Calculate the Vectors Representing Sides AB and AC To use the Angle Bisector Theorem, we first need the vectors representing the sides $AB$ and $AC$. Vector $\vec{AB} = \vec{b} - \vec{a} = (2\widehat i + 3\widehat j + 4\widehat k) - (4\widehat i + 7\widehat j + 8\widehat k)$ $\vec{AB} = (2-4)\widehat i + (3-7)\widehat j + (4-8)\widehat k = -2\widehat i - 4\widehat j - 4\widehat k$ Vector $\vec{AC} = \vec{c} - \vec{a} = (2\widehat i + 5\widehat j + 7\widehat k) - (4\widehat i + 7\widehat j + 8\widehat k)$ $\vec{AC} = (2-4)\widehat i + (5-7)\widehat j + (7-8)\widehat k = -2\widehat i - 2\widehat j - 1\widehat k$

• Step 3: Calculate the Lengths of Sides AB and AC The lengths of the sides are the magnitudes of these vectors. Length of $AB$, denoted by $AB$ or $|\vec{AB}|$: $AB = |\vec{AB}| = \sqrt{(-2)^2 + (-4)^2 + (-4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6$ Length of $AC$, denoted by $AC$ or $|\vec{AC}|$: $AC = |\vec{AC}| = \sqrt{(-2)^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$

• Step 4: Apply the Angle Bisector Theorem to Find the Ratio of Division According to the Angle Bisector Theorem, point $D$ divides the side $BC$ in the ratio $AB:AC$. $\frac{BD}{DC} = \frac{AB}{AC} = \frac{6}{3} = \frac{2}{1}$ Thus, point $D$ divides the line segment $BC$ internally in the ratio $m:n = 2:1$.

• Step 5: Use the Section Formula to Find the Position Vector of D We use the section formula for internal division with $m=2$, $n=1$, $\vec{b} = 2\widehat i + 3\widehat j + 4\widehat k$, and $\vec{c} = 2\widehat i + 5\widehat j + 7\widehat k$. The position vector of $D$, $\vec{d}$, is given by: $\vec{d} = \frac{n\vec{b} + m\vec{c}}{m+n}$ Substituting the values: $\vec{d} = \frac{1 \cdot (2\widehat i + 3\widehat j + 4\widehat k) + 2 \cdot (2\widehat i + 5\widehat j + 7\widehat k)}{2+1}$

• Step 6: Perform Vector Addition and Scalar Multiplication First, perform the scalar multiplication: $1 \cdot (2\widehat i + 3\widehat j + 4\widehat k) = 2\widehat i + 3\widehat j + 4\widehat k$ $2 \cdot (2\widehat i + 5\widehat j + 7\widehat k) = 4\widehat i + 10\widehat j + 14\widehat k$ Now, add these resulting vectors: $(2\widehat i + 3\widehat j + 4\widehat k) + (4\widehat i + 10\widehat j + 14\widehat k) = (2+4)\widehat i + (3+10)\widehat j + (4+14)\widehat k$ $= 6\widehat i + 13\widehat j + 18\widehat k$ Finally, divide by the sum of the ratio ($m+n=3$): $\vec{d} = \frac{6\widehat i + 13\widehat j + 18\widehat k}{3}$ $\vec{d} = \frac{1}{3}(6\widehat i + 13\widehat j + 18\widehat k)$

1. Common Mistakes & Tips

• Incorrect Ratio Assignment: Ensure that the ratio $m:n$ is correctly applied in the section formula. The term $m$ should be multiplied by the position vector of the point that is further from the dividing point (i.e., $\vec{c}$), and $n$ by the position vector of the point closer to the dividing point (i.e., $\vec{b}$), based on the ratio $BD:DC = m:n$.
• Sign Errors in Vector Subtraction: When calculating vectors like $\vec{AB} = \vec{b} - \vec{a}$, pay close attention to the signs of the components.
• Magnitude Calculation Errors: Double-check the squaring and summing of components when calculating the magnitude of vectors.

1. Summary

The problem requires finding the position vector of the point where the angle bisector of $\angle A$ meets the side $BC$. This is achieved by first applying the Angle Bisector Theorem to determine the ratio in which the bisector divides $BC$, using the lengths of sides $AB$ and $AC$. Subsequently, the Section Formula for internal division is used with the calculated ratio and the position vectors of $B$ and $C$ to find the required position vector. The calculated position vector is $\frac{1}{3}(6\widehat i + 13\widehat j + 18\widehat k)$.

The final answer is \boxed{\frac{1}{3}\left( {6\widehat i + 13\widehat j + 18\widehat k} \right)}. This corresponds to option (C).