If the position vectors of points A and B are `3hat(i)-2hat(j)+hat(k) and 2hat(i)+4hat(j)-3hat(k)` respectively, then what is the length of `vec(AB)`?

Options

A. `sqrt(14)`
B. `sqrt(29)`
C. `sqrt(43)`
D. `sqrt(53)`

Explanation

To find the length of the vector \( \vec{AB} \) given the position vectors of points A and B, we can follow these steps:

1. **Identify the position vectors**:
- The position vector of point A is given as \( \vec{A} = 3\hat{i} - 2\hat{j} + \hat{k} \).
- The position vector of point B is given as \( \vec{B} = 2\hat{i} + 4\hat{j} - 3\hat{k} \).

2. **Calculate the vector \( \vec{AB} \)**:
- The vector \( \vec{AB} \) can be calculated using the formula:
\[ \vec{AB} = \vec{B} - \vec{A} \]
- Substituting the position vectors:
\[ \vec{AB} = (2\hat{i} + 4\hat{j} - 3\hat{k}) - (3\hat{i} - 2\hat{j} + \hat{k}) \]

3. **Simplify the expression**:
- Distributing the negative sign:
\[ \vec{AB} = 2\hat{i} + 4\hat{j} - 3\hat{k} - 3\hat{i} + 2\hat{j} - \hat{k} \]
- Combine like terms:
\[ \vec{AB} = (2 - 3)\hat{i} + (4 + 2)\hat{j} + (-3 - 1)\hat{k} \]
\[ \vec{AB} = -1\hat{i} + 6\hat{j} - 4\hat{k} \]

4. **Find the length of \( \vec{AB} \)**:
- The length (magnitude) of the vector \( \vec{AB} \) is given by:
\[ |\vec{AB}| = \sqrt{(-1)^2 + (6)^2 + (-4)^2} \]
- Calculating each term:
\[ |\vec{AB}| = \sqrt{1 + 36 + 16} \]
- Adding these values:
\[ |\vec{AB}| = \sqrt{53} \]

5. **Conclusion**:
- The length of \( \vec{AB} \) is \( \sqrt{53} \).