If `A(veca)=2hati+2hatj+2hatk` & `B(vecb)=2hati+4hatj+4hatk` find the length of angle bisector of `angleAOB`

Options

A. `1/2sqrt(136)`
B. `1/2sqrt(236)`
C. `1/3sqrt(236)`
D. `1/3sqrt(136)`

Answer

To find the length of the angle bisector of angle AOB given the position vectors of points A and B, we can follow these steps:

### Step 1: Identify the position vectors
Given:
- \( \vec{A} = 2\hat{i} + 2\hat{j} + 2\hat{k} \)
- \( \vec{B} = 2\hat{i} + 4\hat{j} + 4\hat{k} \)

### Step 2: Calculate the magnitudes of the vectors
Calculate the magnitude of \( \vec{A} \):
\[ |\vec{A}| = \sqrt{(2)^2 + (2)^2 + (2)^2} = \sqrt{4 + 4 + 4} = \sqrt{12} = 2\sqrt{3} \]

Calculate the magnitude of \( \vec{B} \):
\[ |\vec{B}| = \sqrt{(2)^2 + (4)^2 + (4)^2} = \sqrt{4 + 16 + 16} = \sqrt{36} = 6 \]

### Step 3: Use the angle bisector theorem
According to the angle bisector theorem, if \( D \) is the point on the angle bisector of \( \angle AOB \), then:
\[ \frac{AD}{DB} = \frac{|\vec{A}|}{|\vec{B}|} = \frac{2\sqrt{3}}{6} = \frac{\sqrt{3}}{3} \]
This means \( AD : DB = 1 : 2 \).

### Step 4: Find the position vector of point D
Using the section formula, the position vector of point \( D \) that divides \( AB \) in the ratio \( 1:2 \) is given by:
\[ \vec{D} = \frac{m\vec{B} + n\vec{A}}{m+n} \]
where \( m = 1 \) and \( n = 2 \):
\[ \vec{D} = \frac{1 \cdot (2\hat{i} + 4\hat{j} + 4\hat{k}) + 2 \cdot (2\hat{i} + 2\hat{j} + 2\hat{k})}{1 + 2} \]
Calculating this gives:
\[ \vec{D} = \frac{(2\hat{i} + 4\hat{j} + 4\hat{k}) + (4\hat{i} + 4\hat{j} + 4\hat{k})}{3} = \frac{(6\hat{i} + 8\hat{j} + 8\hat{k})}{3} = 2\hat{i} + \frac{8}{3}\hat{j} + \frac{8}{3}\hat{k} \]

### Step 5: Calculate the magnitude of vector \( \vec{D} \)
Now, we find the length of the angle bisector \( OD \):
\[ |\vec{D}| = \sqrt{\left(2\right)^2 + \left(\frac{8}{3}\right)^2 + \left(\frac{8}{3}\right)^2} \]
Calculating this gives:
\[ |\vec{D}| = \sqrt{4 + \frac{64}{9} + \frac{64}{9}} = \sqrt{4 + \frac{128}{9}} = \sqrt{\frac{36}{9} + \frac{128}{9}} = \sqrt{\frac{164}{9}} = \frac{\sqrt{164}}{3} \]

### Step 6: Simplify the result
Since \( \sqrt{164} = 2\sqrt{41} \):
\[ |\vec{D}| = \frac{2\sqrt{41}}{3} \]

### Final Answer
The length of the angle bisector of angle AOB is:
\[ \frac{2\sqrt{41}}{3} \]