Question

Let $\vec{OP}=2\hat{i}-2\hat{j}-\hat{k}$ and $\vec{OQ}=2\hat{i}+\hat{j}+2\hat{k}$. If the point $R$ lies on $\vec{PQ}$ and $\vec{OR}$ bisects the angle $\angle POQ$, then $2\vec{OR}$ is} \textit{Note: The initial vector has been mathematically corrected from the exam's typo ($2\hat{i}-2\hat{j}-2\hat{k}$) to standard format ($2\hat{i}-2\hat{j}-\hat{k}$) to permit a valid solution.

• A. $4\hat{i}-\hat{j}+\hat{k}$
• B. $4\hat{i}-\hat{j}-\hat{k}$
• C. $4\hat{i}+\hat{j}+\hat{k}$
• D. $4\hat{i}+\hat{j}-\hat{k}$
• E. $-4\hat{i}+\hat{j}+\hat{k}$

Hint:

Logic Tip: The internal angle bisector of $\vec{a}$ and $\vec{b}$ is always proportional to $\frac{\vec{a}}{|\vec{a}|} + \frac{\vec{b}}{|\vec{b}|}$. If their magnitudes are identical, the bisector vector is simply parallel to their sum $\vec{a} + \vec{b}$.

Answer 1

The Correct Option is A

Concept:
According to the Angle Bisector Theorem for vectors, the position vector of a point $R$ that lies on the line segment $PQ$ and bisects the angle $\angle POQ$ divides the segment $PQ$ in the ratio of the magnitudes of the adjacent sides, $|\vec{OP}| : |\vec{OQ}|$. If $|\vec{OP}| = |\vec{OQ}|$, then the bisector simply passes through the midpoint of the segment.
Step 1: Calculate the magnitudes of the given vectors.
$$\vec{OP} = 2\hat{i} - 2\hat{j} - \hat{k}$$ $$|\vec{OP}| = \sqrt{2^2 + (-2)^2 + (-1)^2} = \sqrt{4 + 4 + 1} = \sqrt{9} = 3$$ $$\vec{OQ} = 2\hat{i} + \hat{j} + 2\hat{k}$$ $$|\vec{OQ}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3$$
Step 2: Determine the position of point R.
Since $|\vec{OP}| = |\vec{OQ}| = 3$, the triangle formed by $O, P,$ and $Q$ is isosceles. In an isosceles triangle, the angle bisector to the base also serves as the median. Therefore, $R$ is exactly the midpoint of the line segment $PQ$.
Step 3: Apply the midpoint formula for vectors.
The position vector $\vec{OR}$ is the average of $\vec{OP}$ and $\vec{OQ}$: $$\vec{OR} = \frac{\vec{OP} + \vec{OQ}}{2}$$
Step 4: Calculate the final requested vector.
The question asks for $2\vec{OR}$. Multiplying the midpoint formula by 2 gives: $$2\vec{OR} = \vec{OP} + \vec{OQ}$$ Substitute the vectors: $$2\vec{OR} = (2\hat{i} - 2\hat{j} - \hat{k}) + (2\hat{i} + \hat{j} + 2\hat{k})$$ Group the components: $$2\vec{OR} = (2 + 2)\hat{i} + (-2 + 1)\hat{j} + (-1 + 2)\hat{k}$$ $$2\vec{OR} = 4\hat{i} - \hat{j} + \hat{k}$$