The perimeter of the triangle whose vertices have the position vectors `hat(i)+hat(j)+hat(k), 5hat(i)+3hat(j)-3hat(k) and 2hat(i)+5hat(j)+9hat(k)` is

Options

A. `15+sqrt(157)`
B. `16+sqrt(157)`
C. `15-sqrt(157)`
D. `sqrt(15)-sqrt(157)`

Answer

To find the perimeter of the triangle whose vertices have the position vectors \(\hat{i} + \hat{j} + \hat{k}\), \(5\hat{i} + 3\hat{j} - 3\hat{k}\), and \(2\hat{i} + 5\hat{j} + 9\hat{k}\), we will follow these steps:

### Step 1: Define the position vectors
Let:
- \( \mathbf{A} = \hat{i} + \hat{j} + \hat{k} \)
- \( \mathbf{B} = 5\hat{i} + 3\hat{j} - 3\hat{k} \)
- \( \mathbf{C} = 2\hat{i} + 5\hat{j} + 9\hat{k} \)

### Step 2: Calculate the vectors representing the sides of the triangle
1. **Vector \( \mathbf{AB} \)**:
\[ \mathbf{AB} = \mathbf{B} - \mathbf{A} = (5\hat{i} + 3\hat{j} - 3\hat{k}) - (\hat{i} + \hat{j} + \hat{k}) \]
\[ = (5 - 1)\hat{i} + (3 - 1)\hat{j} + (-3 - 1)\hat{k} = 4\hat{i} + 2\hat{j} - 4\hat{k} \]

2. **Vector \( \mathbf{BC} \)**:
\[ \mathbf{BC} = \mathbf{C} - \mathbf{B} = (2\hat{i} + 5\hat{j} + 9\hat{k}) - (5\hat{i} + 3\hat{j} - 3\hat{k}) \]
\[ = (2 - 5)\hat{i} + (5 - 3)\hat{j} + (9 + 3)\hat{k} = -3\hat{i} + 2\hat{j} + 12\hat{k} \]

3. **Vector \( \mathbf{CA} \)**:
\[ \mathbf{CA} = \mathbf{A} - \mathbf{C} = (\hat{i} + \hat{j} + \hat{k}) - (2\hat{i} + 5\hat{j} + 9\hat{k}) \]
\[ = (1 - 2)\hat{i} + (1 - 5)\hat{j} + (1 - 9)\hat{k} = -\hat{i} - 4\hat{j} - 8\hat{k} \]

### Step 3: Calculate the magnitudes of the vectors
1. **Magnitude of \( \mathbf{AB} \)**:
\[ |\mathbf{AB}| = \sqrt{(4)^2 + (2)^2 + (-4)^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \]

2. **Magnitude of \( \mathbf{BC} \)**:
\[ |\mathbf{BC}| = \sqrt{(-3)^2 + (2)^2 + (12)^2} = \sqrt{9 + 4 + 144} = \sqrt{157} \]

3. **Magnitude of \( \mathbf{CA} \)**:
\[ |\mathbf{CA}| = \sqrt{(-1)^2 + (-4)^2 + (-8)^2} = \sqrt{1 + 16 + 64} = \sqrt{81} = 9 \]

### Step 4: Calculate the perimeter of the triangle
The perimeter \( P \) is given by the sum of the lengths of the sides:
\[ P = |\mathbf{AB}| + |\mathbf{BC}| + |\mathbf{CA}| = 6 + \sqrt{157} + 9 \]
\[ P = 15 + \sqrt{157} \]

### Final Answer
The perimeter of the triangle is \( 15 + \sqrt{157} \).

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