If `bar(OA)=overset(^)i+3overset(^)j-2overset(^)k and bar(OB)=3overset(^)i+overset(^)j-2overset(^)k,` then the vectors `bar(OC)` which bisects `angleAOB` is equal to

Options

A. `overset(^)i-overset(^)j-overset(^)k`
B. `2(overset(^)i+overset(^)j+overset(^)k)`
C. `-overset(^)i+overset(^)j-overset(^)k`
D. `2(overset(^)i+overset(^)j-overset(^)k)`

Answer

To find the vector \( \overrightarrow{OC} \) that bisects the angle \( AOB \), we can follow these steps:

### Step 1: Identify the given vectors
We are given:
\[ \overrightarrow{OA} = \hat{i} + 3\hat{j} - 2\hat{k} \]
\[ \overrightarrow{OB} = 3\hat{i} + \hat{j} - 2\hat{k} \]

### Step 2: Use the formula for the angle bisector
The position vector of the point \( C \) that bisects the angle \( AOB \) can be found using the formula:
\[ \overrightarrow{OC} = \frac{|\overrightarrow{OB}| \cdot \overrightarrow{OA} + |\overrightarrow{OA}| \cdot \overrightarrow{OB}}{|\overrightarrow{OA}| + |\overrightarrow{OB}|} \]
where \( |\overrightarrow{OA}| \) and \( |\overrightarrow{OB}| \) are the magnitudes of the vectors \( \overrightarrow{OA} \) and \( \overrightarrow{OB} \) respectively.

### Step 3: Calculate the magnitudes of the vectors
First, we calculate the magnitudes:
\[ |\overrightarrow{OA}| = \sqrt{(1)^2 + (3)^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]
\[ |\overrightarrow{OB}| = \sqrt{(3)^2 + (1)^2 + (-2)^2} = \sqrt{9 + 1 + 4} = \sqrt{14} \]

### Step 4: Substitute the values into the angle bisector formula
Now substituting the values into the angle bisector formula:
\[ \overrightarrow{OC} = \frac{|\overrightarrow{OB}| \cdot \overrightarrow{OA} + |\overrightarrow{OA}| \cdot \overrightarrow{OB}}{|\overrightarrow{OA}| + |\overrightarrow{OB}|} \]
\[ = \frac{\sqrt{14} \cdot (\hat{i} + 3\hat{j} - 2\hat{k}) + \sqrt{14} \cdot (3\hat{i} + \hat{j} - 2\hat{k})}{\sqrt{14} + \sqrt{14}} \]
\[ = \frac{\sqrt{14}(\hat{i} + 3\hat{j} - 2\hat{k} + 3\hat{i} + \hat{j} - 2\hat{k})}{2\sqrt{14}} \]
\[ = \frac{\sqrt{14}(4\hat{i} + 4\hat{j} - 4\hat{k})}{2\sqrt{14}} \]
\[ = \frac{4(\hat{i} + \hat{j} - \hat{k})}{2} \]
\[ = 2(\hat{i} + \hat{j} - \hat{k}) \]

### Step 5: Final answer
Thus, the vector \( \overrightarrow{OC} \) that bisects the angle \( AOB \) is:
\[ \overrightarrow{OC} = 2\hat{i} + 2\hat{j} - 2\hat{k} \]