Find the perimeter of a triangle with sides `3hati+4hatj+5hatk,4hati-3hatj-5hatk and 7 hati+hatj`.

Answer

To find the perimeter of the triangle formed by the vectors \( \mathbf{A} = 3\hat{i} + 4\hat{j} + 5\hat{k} \), \( \mathbf{B} = 4\hat{i} - 3\hat{j} - 5\hat{k} \), and \( \mathbf{C} = 7\hat{i} + \hat{j} \), we will follow these steps:

### Step 1: Identify the vectors
Let:
- \( \mathbf{A} = 3\hat{i} + 4\hat{j} + 5\hat{k} \)
- \( \mathbf{B} = 4\hat{i} - 3\hat{j} - 5\hat{k} \)
- \( \mathbf{C} = 7\hat{i} + \hat{j} \)

### Step 2: Calculate the lengths of the sides of the triangle
To find the perimeter, we need to calculate the lengths of the sides represented by these vectors.

1. **Length of side AB**:
\[ \text{AB} = |\mathbf{A} - \mathbf{B}| \]
\[ \mathbf{A} - \mathbf{B} = (3\hat{i} + 4\hat{j} + 5\hat{k}) - (4\hat{i} - 3\hat{j} - 5\hat{k}) = (3 - 4)\hat{i} + (4 + 3)\hat{j} + (5 + 5)\hat{k} = -\hat{i} + 7\hat{j} + 10\hat{k} \]
\[ |\mathbf{A} - \mathbf{B}| = \sqrt{(-1)^2 + 7^2 + 10^2} = \sqrt{1 + 49 + 100} = \sqrt{150} \]

2. **Length of side BC**:
\[ \text{BC} = |\mathbf{B} - \mathbf{C}| \]
\[ \mathbf{B} - \mathbf{C} = (4\hat{i} - 3\hat{j} - 5\hat{k}) - (7\hat{i} + \hat{j}) = (4 - 7)\hat{i} + (-3 - 1)\hat{j} + (-5 - 0)\hat{k} = -3\hat{i} - 4\hat{j} - 5\hat{k} \]
\[ |\mathbf{B} - \mathbf{C}| = \sqrt{(-3)^2 + (-4)^2 + (-5)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]

3. **Length of side CA**:
\[ \text{CA} = |\mathbf{C} - \mathbf{A}| \]
\[ \mathbf{C} - \mathbf{A} = (7\hat{i} + \hat{j}) - (3\hat{i} + 4\hat{j} + 5\hat{k}) = (7 - 3)\hat{i} + (1 - 4)\hat{j} + (0 - 5)\hat{k} = 4\hat{i} - 3\hat{j} - 5\hat{k} \]
\[ |\mathbf{C} - \mathbf{A}| = \sqrt{4^2 + (-3)^2 + (-5)^2} = \sqrt{16 + 9 + 25} = \sqrt{50} \]

### Step 3: Calculate the perimeter of the triangle
The perimeter \( P \) of the triangle is given by the sum of the lengths of its sides:
\[ P = |\mathbf{A} - \mathbf{B}| + |\mathbf{B} - \mathbf{C}| + |\mathbf{C} - \mathbf{A}| \]
\[ P = \sqrt{150} + \sqrt{50} + \sqrt{50} \]
\[ P = \sqrt{150} + 2\sqrt{50} \]

### Step 4: Simplify the perimeter
We can simplify \( \sqrt{150} \) and \( \sqrt{50} \):
\[ \sqrt{150} = \sqrt{25 \times 6} = 5\sqrt{6} \]
\[ \sqrt{50} = \sqrt{25 \times 2} = 5\sqrt{2} \]
Thus,
\[ P = 5\sqrt{6} + 2(5\sqrt{2}) = 5\sqrt{6} + 10\sqrt{2} \]

### Final Answer
The perimeter of the triangle is \( 5\sqrt{6} + 10\sqrt{2} \).

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