Let `vec(OA)=hati+3hatj-2hatk and vec(OB)=3hati+hatj-2hatk.` Then vector `vec(OC)` biecting the angle `AOB and C` being a point on the line `AB` is

Options

A. `4(hati + hatj - hatk)`
B. `2(hati + hatj - hatk)`
C. `hati + hatj - hatk`
D. none of these

Explanation

To solve the problem, we need to find the vector \(\vec{OC}\) that bisects the angle between the vectors \(\vec{OA}\) and \(\vec{OB}\). The vectors are given as:

\[ \vec{OA} = \hat{i} + 3\hat{j} - 2\hat{k} \]
\[ \vec{OB} = 3\hat{i} + \hat{j} - 2\hat{k} \]

### Step 1: Calculate the magnitudes of \(\vec{OA}\) and \(\vec{OB}\)

The magnitude of a vector \(\vec{v} = a\hat{i} + b\hat{j} + c\hat{k}\) is given by:

\[ |\vec{v}| = \sqrt{a^2 + b^2 + c^2} \]

Calculating the magnitude of \(\vec{OA}\):

\[ |\vec{OA}| = \sqrt{1^2 + 3^2 + (-2)^2} = \sqrt{1 + 9 + 4} = \sqrt{14} \]

Calculating the magnitude of \(\vec{OB}\):

\[ |\vec{OB}| = \sqrt{3^2 + 1^2 + (-2)^2} = \sqrt{9 + 1 + 4} = \sqrt{14} \]

### Step 2: Find the unit vectors of \(\vec{OA}\) and \(\vec{OB}\)

The unit vector \(\hat{u}\) of a vector \(\vec{v}\) is given by:

\[ \hat{u} = \frac{\vec{v}}{|\vec{v}|} \]

Finding the unit vector of \(\vec{OA}\):

\[ \hat{u_A} = \frac{\vec{OA}}{|\vec{OA}|} = \frac{\hat{i} + 3\hat{j} - 2\hat{k}}{\sqrt{14}} = \frac{1}{\sqrt{14}}\hat{i} + \frac{3}{\sqrt{14}}\hat{j} - \frac{2}{\sqrt{14}}\hat{k} \]

Finding the unit vector of \(\vec{OB}\):

\[ \hat{u_B} = \frac{\vec{OB}}{|\vec{OB}|} = \frac{3\hat{i} + \hat{j} - 2\hat{k}}{\sqrt{14}} = \frac{3}{\sqrt{14}}\hat{i} + \frac{1}{\sqrt{14}}\hat{j} - \frac{2}{\sqrt{14}}\hat{k} \]

### Step 3: Calculate the angle bisector vector \(\vec{OC}\)

The angle bisector vector \(\vec{OC}\) can be calculated using the formula:

\[ \vec{OC} = \frac{|\vec{OB}| \cdot \vec{OA} + |\vec{OA}| \cdot \vec{OB}}{|\vec{OA}| + |\vec{OB}|} \]

Since both magnitudes are equal (\(|\vec{OA}| = |\vec{OB}| = \sqrt{14}\)), we can simplify this to:

\[ \vec{OC} = \frac{\vec{OA} + \vec{OB}}{2} \]

Calculating \(\vec{OC}\):

\[ \vec{OC} = \frac{(\hat{i} + 3\hat{j} - 2\hat{k}) + (3\hat{i} + \hat{j} - 2\hat{k})}{2} \]

\[ = \frac{(1 + 3)\hat{i} + (3 + 1)\hat{j} + (-2 - 2)\hat{k}}{2} \]

\[ = \frac{4\hat{i} + 4\hat{j} - 4\hat{k}}{2} \]

\[ = 2\hat{i} + 2\hat{j} - 2\hat{k} \]

### Final Answer

Thus, the vector \(\vec{OC}\) is:

\[ \vec{OC} = 2\hat{i} + 2\hat{j} - 2\hat{k} \]