The position vectors of A and B are `2hati-9hatj-4hatk and 6hati-3hatj+8hatk` respectively, then the magnitude of AB is

Options

A. 11
B. 12
C. 13
D. 14

Explanation

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

### Step 1: Write down the position vectors
The position vectors of points A and B are given as:
- \( \overrightarrow{A} = 2\hat{i} - 9\hat{j} - 4\hat{k} \)
- \( \overrightarrow{B} = 6\hat{i} - 3\hat{j} + 8\hat{k} \)

### Step 2: Find the vector \( \overrightarrow{AB} \)
The vector \( \overrightarrow{AB} \) can be calculated using the formula:
\[ \overrightarrow{AB} = \overrightarrow{B} - \overrightarrow{A} \]
Substituting the position vectors:
\[ \overrightarrow{AB} = (6\hat{i} - 3\hat{j} + 8\hat{k}) - (2\hat{i} - 9\hat{j} - 4\hat{k}) \]

### Step 3: Simplify the expression
Now, we simplify the expression:
\[ \overrightarrow{AB} = (6 - 2)\hat{i} + (-3 + 9)\hat{j} + (8 + 4)\hat{k} \]
Calculating each component:
\[ \overrightarrow{AB} = 4\hat{i} + 6\hat{j} + 12\hat{k} \]

### Step 4: Find the magnitude of \( \overrightarrow{AB} \)
The magnitude of a vector \( \overrightarrow{V} = a\hat{i} + b\hat{j} + c\hat{k} \) is given by:
\[ |\overrightarrow{V}| = \sqrt{a^2 + b^2 + c^2} \]
For \( \overrightarrow{AB} = 4\hat{i} + 6\hat{j} + 12\hat{k} \):
\[ |\overrightarrow{AB}| = \sqrt{4^2 + 6^2 + 12^2} \]
Calculating each term:
\[ |\overrightarrow{AB}| = \sqrt{16 + 36 + 144} \]
\[ |\overrightarrow{AB}| = \sqrt{196} \]
\[ |\overrightarrow{AB}| = 14 \]

### Final Answer
The magnitude of \( \overrightarrow{AB} \) is \( 14 \).

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