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

Options

A. `sqrt14`
B. `sqrt29`
C. `sqrt43`
D. `sqrt53`

Answer

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

### Step 1: Identify the position vectors
The position vectors of points A and B are given as:
\[ \vec{OA} = 3\hat{i} - 2\hat{j} + \hat{k} \]
\[ \vec{OB} = 2\hat{i} + 4\hat{j} - 3\hat{k} \]

### Step 2: Calculate the vector \(\vec{AB}\)
The vector \(\vec{AB}\) can be calculated using the formula:
\[ \vec{AB} = \vec{OB} - \vec{OA} \]
Substituting the values:
\[ \vec{AB} = (2\hat{i} + 4\hat{j} - 3\hat{k}) - (3\hat{i} - 2\hat{j} + \hat{k}) \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ \vec{AB} = 2\hat{i} + 4\hat{j} - 3\hat{k} - 3\hat{i} + 2\hat{j} - \hat{k} \]
Combining like terms:
\[ \vec{AB} = (2 - 3)\hat{i} + (4 + 2)\hat{j} + (-3 - 1)\hat{k} \]
\[ \vec{AB} = -\hat{i} + 6\hat{j} - 4\hat{k} \]

### Step 4: Calculate the length of \(\vec{AB}\)
The length of the vector \(\vec{AB}\) is given by the formula:
\[ |\vec{AB}| = \sqrt{(-1)^2 + (6)^2 + (-4)^2} \]
Calculating each term:
\[ |\vec{AB}| = \sqrt{1 + 36 + 16} \]
\[ |\vec{AB}| = \sqrt{53} \]

### Final Answer
Thus, the length of \(\vec{AB}\) is:
\[ |\vec{AB}| = \sqrt{53} \]