The perimeter of the triangle with sides `3hat(i)+4hat(j)+5hat(k), 4hat(i)-3hat(j)+5hat(k) and 7hat(i)+hat(j)` is

Options

A. `sqrt(450)`
B. `sqrt(150)`
C. `sqrt(50)`
D. `sqrt(200)`

Answer

To find the perimeter of the triangle with sides represented 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 \( \mathbf{C} = 7\hat{i} + \hat{j} + 0\hat{k} \), we will follow these steps:

### Step 1: Identify the position vectors
We have the following position vectors:
- \( \mathbf{A} = (3, 4, 5) \)
- \( \mathbf{B} = (4, -3, 5) \)
- \( \mathbf{C} = (7, 1, 0) \)

### Step 2: Calculate the lengths of the sides of the triangle
We will use the distance formula to find the lengths of the sides \( AB \), \( BC \), and \( CA \).

#### Length of side \( AB \):
Using the distance formula:
\[ AB = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Substituting the coordinates of points \( A \) and \( B \):
\[ AB = \sqrt{(4 - 3)^2 + (-3 - 4)^2 + (5 - 5)^2} = \sqrt{1^2 + (-7)^2 + 0^2} = \sqrt{1 + 49 + 0} = \sqrt{50} \]

#### Length of side \( BC \):
Using the distance formula again:
\[ BC = \sqrt{(7 - 4)^2 + (1 - (-3))^2 + (0 - 5)^2} \]
Calculating:
\[ BC = \sqrt{(3)^2 + (4)^2 + (-5)^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]

#### Length of side \( CA \):
Using the distance formula:
\[ CA = \sqrt{(7 - 3)^2 + (1 - 4)^2 + (0 - 5)^2} \]
Calculating:
\[ CA = \sqrt{(4)^2 + (-3)^2 + (-5)^2} = \sqrt{16 + 9 + 25} = \sqrt{50} \]

### Step 3: Calculate the perimeter of the triangle
The perimeter \( P \) of triangle \( ABC \) is given by:
\[ P = AB + BC + CA \]
Substituting the lengths we found:
\[ P = \sqrt{50} + \sqrt{50} + \sqrt{50} = 3\sqrt{50} \]

### Step 4: Simplify the perimeter
We can simplify \( 3\sqrt{50} \):
\[ 3\sqrt{50} = 3\sqrt{25 \cdot 2} = 3 \cdot 5\sqrt{2} = 15\sqrt{2} \]

### Final Answer
Thus, the perimeter of the triangle is:
\[ \boxed{15\sqrt{2}} \]