1. Key Concepts and Formulas

• Position Vectors and Magnitude: The position vector of a point $P$ from the origin $O$ is $\vec{OP}$. Its magnitude (length) is $|\vec{OP}| = \sqrt{x^2+y^2+z^2}$ for $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$.
• Angle Bisector Theorem (Vector Form): If the internal bisector of $\angle AOB$ meets the line segment $AB$ at point $C$, then $C$ divides $AB$ internally in the ratio of the lengths of the adjacent sides, i.e., $AC:CB = OA:OB$.
• Section Formula (Vector Form): If a point $C$ divides the line segment joining points $A$ and $B$ with position vectors $\vec{OA}$ and $\vec{OB}$ respectively, in the ratio $m:n$ internally, then the position vector of $C$ is given by: $\vec{OC} = \frac{n \vec{OA} + m \vec{OB}}{m+n}$

2. Step-by-Step Solution

Step 1: Calculate the magnitudes of vectors $\vec{OA}$ and $\vec{OB}$.

• Why this step? The Angle Bisector Theorem requires the lengths of the sides $OA$ and $OB$ to determine the ratio in which point $C$ divides the line segment $AB$.

• Calculation: Given position vector of $A$: $\vec{OA} = 2 \hat{i}+2 \hat{j}+\hat{k}$ $|\vec{OA}| = \sqrt{(2)^2 + (2)^2 + (1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$ Given position vector of $B$: $\vec{OB} = 2 \hat{i}+4 \hat{j}+4 \hat{k}$ $|\vec{OB}| = \sqrt{(2)^2 + (4)^2 + (4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6$

Step 2: Determine the ratio in which $C$ divides the line segment $AB$.

• Why this step? According to the Angle Bisector Theorem, point $C$ divides $AB$ internally in the ratio $OA:OB$. This ratio will be used in the section formula.

• Calculation: The ratio $OA:OB = |\vec{OA}|:|\vec{OB}| = 3:6 = 1:2$. So, $C$ divides the line segment $AB$ internally in the ratio $1:2$. This means for the section formula, we take $m=1$ and $n=2$ (where $C$ divides $AB$ in ratio $m:n$).

Step 3: Find the position vector of point $C$ ($\vec{OC}$) using the section formula.

• Why this step? With the ratio of division determined, we can apply the section formula to find the coordinates (or position vector) of point $C$.

• Calculation: Using the section formula $\vec{OC} = \frac{n \vec{OA} + m \vec{OB}}{m+n}$ with $m=1$, $n=2$, $\vec{OA} = 2 \hat{i}+2 \hat{j}+\hat{k}$, and $\vec{OB} = 2 \hat{i}+4 \hat{j}+4 \hat{k}$: $\vec{OC} = \frac{2 \cdot (2 \hat{i}+2 \hat{j}+\hat{k}) + 1 \cdot (2 \hat{i}+4 \hat{j}+4 \hat{k})}{1+2}$ $\vec{OC} = \frac{(4 \hat{i}+4 \hat{j}+2 \hat{k}) + (2 \hat{i}+4 \hat{j}+4 \hat{k})}{3}$ Combine the components: $\vec{OC} = \frac{(4+2)\hat{i} + (4+4)\hat{j} + (2+4)\hat{k}}{3}$ $\vec{OC} = \frac{6\hat{i} + 8\hat{j} + 6\hat{k}}{3}$ $\vec{OC} = 2\hat{i} + \frac{8}{3}\hat{j} + 2\hat{k}$

Step 4: Calculate the length of $OC$.

• Why this step? The problem asks for the length of $OC$, which is the magnitude of the position vector $\vec{OC}$ obtained in the previous step.

• Calculation: $|\vec{OC}| = \sqrt{(2)^2 + \left(\frac{8}{3}\right)^2 + (2)^2}$ $|\vec{OC}| = \sqrt{4 + \frac{64}{9} + 4}$ $|\vec{OC}| = \sqrt{8 + \frac{64}{9}}$ To add these terms, find a common denominator: $|\vec{OC}| = \sqrt{\frac{8 \times 9}{9} + \frac{64}{9}} = \sqrt{\frac{72}{9} + \frac{64}{9}} = \sqrt{\frac{136}{9}}$ Separate the square root for the numerator and denominator: $|\vec{OC}| = \frac{\sqrt{136}}{\sqrt{9}} = \frac{\sqrt{4 \times 34}}{3}$ Simplify the radical: $|\vec{OC}| = \frac{2\sqrt{34}}{3}$

3. Common Mistakes & Tips

• Incorrect Ratio for Section Formula: A common mistake is to use the magnitudes in the wrong order for the section formula. Remember, if $C$ divides $AB$ in ratio $m:n$, then $\vec{OC} = \frac{n\vec{OA} + m\vec{OB}}{m+n}$. The Angle Bisector Theorem states $AC:CB = OA:OB$, so if $AC:CB = m:n$, then $m=OA$ and $n=OB$.
• Vector vs. Scalar Operations: Be careful to distinguish between magnitudes (scalars) and position vectors. The Angle Bisector Theorem uses ratios of lengths (scalars), not vectors.
• Simplification of Radicals: Always simplify square roots to their simplest form. For example, $\sqrt{136} = \sqrt{4 \times 34} = 2\sqrt{34}$.

4. Summary

The problem involves applying the Angle Bisector Theorem in a 3D vector context. First, we calculated the magnitudes of the position vectors $\vec{OA}$ and $\vec{OB}$. These magnitudes determined the ratio in which the angle bisector $OC$ divides the line segment $AB$. Using the section formula, we found the position vector of point $C$. Finally, the length of $OC$ was calculated by finding the magnitude of $\vec{OC}$. Following these steps, the calculated length of $OC$ is $\frac{2}{3}\sqrt{34}$.

5. Final Answer

The final answer is $\boxed{\frac{2}{3} \sqrt{34}}$, which corresponds to option (C).