The position vectors of the vertices `A,` `B` and `C` of a triangle are `2 hat i-3 hat j+3 hat k,` `2 hat i+2 hat j+3 hat k` and ,`-hat i+hat j+3 hat k` respectively. Let `l` denotes the length of the angle bisector `AD` of `/_BAC` where `D` is on the line segment `BC`, then `2l^(2)`` equals:

Options

A. 50
B. 45
C. 42
D. 49

Answer

To solve the problem, we need to find the length of the angle bisector \( AD \) of angle \( \angle BAC \) in triangle \( ABC \) where the vertices are given by their position vectors.

1. **Identify the coordinates of points A, B, and C from the position vectors:**
- The position vector of point \( A \) is \( \vec{A} = 2\hat{i} - 3\hat{j} + 3\hat{k} \) which corresponds to the coordinates \( A(2, -3, 3) \).
- The position vector of point \( B \) is \( \vec{B} = 2\hat{i} + 2\hat{j} + 3\hat{k} \) which corresponds to the coordinates \( B(2, 2, 3) \).
- The position vector of point \( C \) is \( \vec{C} = -\hat{i} + \hat{j} + 3\hat{k} \) which corresponds to the coordinates \( C(-1, 1, 3) \).

2. **Find the coordinates of point D, the point on segment BC:**
- The coordinates of point \( D \) can be found using the section formula. Since \( D \) is on segment \( BC \), we need to find the ratio in which \( D \) divides \( BC \).
- The coordinates of \( D \) can be calculated as:
\[ D = \left( \frac{x_B + x_C}{2}, \frac{y_B + y_C}{2}, z_B \right) = \left( \frac{2 + (-1)}{2}, \frac{2 + 1}{2}, 3 \right) = \left( \frac{1}{2}, \frac{3}{2}, 3 \right) \]

3. **Calculate the length of the angle bisector \( AD \):**
- The length \( l \) of the angle bisector \( AD \) can be calculated using the distance formula:
\[ l = AD = \sqrt{(x_D - x_A)^2 + (y_D - y_A)^2 + (z_D - z_A)^2} \]
- Plugging in the coordinates:
\[ l = \sqrt{\left(\frac{1}{2} - 2\right)^2 + \left(\frac{3}{2} + 3\right)^2 + (3 - 3)^2} \]
- Simplifying:
\[ = \sqrt{\left(-\frac{3}{2}\right)^2 + \left(\frac{9}{2}\right)^2 + 0^2} \]
\[ = \sqrt{\frac{9}{4} + \frac{81}{4}} = \sqrt{\frac{90}{4}} = \sqrt{\frac{45}{2}} = \frac{3\sqrt{10}}{2} \]

4. **Calculate \( 2l^2 \):**
- Now we calculate \( 2l^2 \):
\[ l^2 = \left(\frac{3\sqrt{10}}{2}\right)^2 = \frac{9 \cdot 10}{4} = \frac{90}{4} = \frac{45}{2} \]
\[ 2l^2 = 2 \cdot \frac{45}{2} = 45 \]

Thus, the final answer is:
\[ \boxed{45} \]