Question

The line integral of the vector field \[ \mathbf{F} = zx \hat{i} + xy \hat{j} + yz \hat{k} \] along the boundary of the triangle with vertices \( (1,0,0), (0,1,0) \) and \( (0,0,1) \), oriented anti-clockwise, when viewed from the point \( (2,2,2) \), is \[ \int_C \mathbf{F} \cdot d\mathbf{r} = \]

• A. \( -\frac{1}{2} \)
• B. 1
• C. \( \frac{1}{2} \)
• D. 2

Hint:

For line integrals over a closed path, apply Stokes' Theorem when possible to simplify the calculation.

Answer 1

The Correct Option is B

Step 1: Parametrize the path.
The line integral is taken along the boundary of the triangle with the given vertices. Parametrize each segment of the boundary.
Step 2: Apply Stokes’ Theorem.
By using Stokes' Theorem and the vector field \( \mathbf{F} \), we find that the line integral is 1.
Step 3: Conclusion.
Thus, the correct answer is (B).