Three vectors `vec(A) = 2hat(i) - hat(j) + hat(k), vec(B) = hat(i) - 3hat(j) - 5hat(k)`, and `vec(C ) = 3hat(i) - 4hat(j) - 4hat(k)` are sides of an :

Options

A. equilateral triangle
B. right angled triangle
C. isosceles triangle
D. none of these

Answer

To determine if the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) represent the sides of a triangle, we will first calculate the magnitudes of each vector and then check if they satisfy the conditions for a right triangle.

### Step 1: Calculate the magnitude of \(\vec{A}\)
Given:
\[ \vec{A} = 2\hat{i} - \hat{j} + \hat{k} \]
The magnitude of \(\vec{A}\) is calculated as:
\[ |\vec{A}| = \sqrt{(2)^2 + (-1)^2 + (1)^2} = \sqrt{4 + 1 + 1} = \sqrt{6} \]

### Step 2: Calculate the magnitude of \(\vec{B}\)
Given:
\[ \vec{B} = \hat{i} - 3\hat{j} - 5\hat{k} \]
The magnitude of \(\vec{B}\) is calculated as:
\[ |\vec{B}| = \sqrt{(1)^2 + (-3)^2 + (-5)^2} = \sqrt{1 + 9 + 25} = \sqrt{35} \]

### Step 3: Calculate the magnitude of \(\vec{C}\)
Given:
\[ \vec{C} = 3\hat{i} - 4\hat{j} - 4\hat{k} \]
The magnitude of \(\vec{C}\) is calculated as:
\[ |\vec{C}| = \sqrt{(3)^2 + (-4)^2 + (-4)^2} = \sqrt{9 + 16 + 16} = \sqrt{41} \]

### Step 4: Check if the triangle is a right triangle
To check if the triangle formed by these vectors is a right triangle, we need to see if the Pythagorean theorem holds. The largest magnitude will be considered as the hypotenuse.

1. Identify the largest magnitude:
- \(|\vec{C}| = \sqrt{41}\) (largest)
- \(|\vec{B}| = \sqrt{35}\)
- \(|\vec{A}| = \sqrt{6}\)

2. Check the Pythagorean theorem:
\[ |\vec{C}|^2 = |\vec{A}|^2 + |\vec{B}|^2 \]
Substituting the values:
\[ 41 = 6 + 35 \]
\[ 41 = 41 \quad \text{(True)} \]

### Conclusion
Since the Pythagorean theorem holds, the vectors \(\vec{A}\), \(\vec{B}\), and \(\vec{C}\) represent the sides of a right triangle.