The position vectors of the points A and B are respectively `3hat(i)-5hat(5) + 2 hat(k) and hat(i)+hat(j)-hat(k)`. What is the length of AB ?

Options

A. 11
B. 9
C. 7
D. 6

Explanation

To find the length of the line segment 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 follows:
- Position vector of A: **OA** = \(3\hat{i} - 5\hat{j} + 2\hat{k}\)
- Position vector of B: **OB** = \(\hat{i} + \hat{j} - \hat{k}\)

### Step 2: Write the vectors in component form
We can express the position vectors in component form:
- **A** = (3, -5, 2)
- **B** = (1, 1, -1)

### Step 3: Find the vector AB
The vector **AB** can be found using the formula:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Calculating this gives:
\[ \vec{AB} = (1 - 3)\hat{i} + (1 - (-5))\hat{j} + (-1 - 2)\hat{k} \]
\[ \vec{AB} = -2\hat{i} + 6\hat{j} - 3\hat{k} \]

### Step 4: Calculate the magnitude of vector AB
The length of vector **AB** can be calculated using the formula for the magnitude of a vector:
\[ |\vec{AB}| = \sqrt{(-2)^2 + 6^2 + (-3)^2} \]
Calculating this gives:
\[ |\vec{AB}| = \sqrt{4 + 36 + 9} \]
\[ |\vec{AB}| = \sqrt{49} \]
\[ |\vec{AB}| = 7 \]

### Conclusion
The length of AB is **7 units**.

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