If vectors `vec(A)=(hat(i)+2hat(j)+3hat(k))m` and `vec(B)=(hat(i)-hat(j)-hat(k))m` represent two sides of a triangle, then the third side can have length equal to `:`

Options

A. `3m`
B. `4m`
C. `6m`
D. Can not be determined

Explanation

To find the length of the third side of the triangle formed by the vectors \(\vec{A}\) and \(\vec{B}\), we can follow these steps:

### Step 1: Write down the vectors
Given:
\[ \vec{A} = \hat{i} + 2\hat{j} + 3\hat{k} \quad \text{(in meters)} \]
\[ \vec{B} = \hat{i} - \hat{j} - \hat{k} \quad \text{(in meters)} \]

### Step 2: Find the resultant vector (third side)
The third side of the triangle can be found by adding the two vectors:
\[ \vec{C} = \vec{A} + \vec{B} \]
Calculating the components:
\[ \vec{C} = (\hat{i} + 2\hat{j} + 3\hat{k}) + (\hat{i} - \hat{j} - \hat{k}) \]
Combine the like terms:
\[ \vec{C} = (1 + 1)\hat{i} + (2 - 1)\hat{j} + (3 - 1)\hat{k} \]
\[ \vec{C} = 2\hat{i} + 1\hat{j} + 2\hat{k} \]

### Step 3: Calculate the magnitude of the resultant vector
The magnitude of vector \(\vec{C}\) is given by:
\[ |\vec{C}| = \sqrt{(2)^2 + (1)^2 + (2)^2} \]
Calculating each term:
\[ |\vec{C}| = \sqrt{4 + 1 + 4} = \sqrt{9} \]
Thus,
\[ |\vec{C}| = 3 \text{ meters} \]

### Conclusion
The length of the third side of the triangle is:
\[ \boxed{3 \text{ meters}} \]

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