The following sets of three vectors act on a body. Whose resultant cannot be zero ?

Options

A. 10, 10, 10
B. 10, 10, 20
C. 10, 20, 20
D. 10, 20, 40

Explanation

To determine which set of three vectors cannot have a resultant of zero, we can analyze the conditions under which three vectors can sum to zero. The key concept here is the triangle inequality, which states that for any three sides of a triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

### Step-by-Step Solution:

1. **Understanding the Problem**: We need to identify which set of three vectors cannot result in a zero vector when added together. This can be determined by checking the triangle inequality conditions.

2. **Triangle Inequality Conditions**:
- For three vectors A, B, and C, they can form a closed triangle (resultant = 0) if:
- \( A + B > C \)
- \( A + C > B \)
- \( B + C > A \)

3. **Analyzing the Options**:
- **Option A**: Vectors can form an equilateral triangle. Here, all sides are equal, so the conditions are satisfied. Resultant can be zero.
- **Option B**: If \( A + B = C \), they are collinear and can sum to zero. Resultant can be zero.
- **Option C**: If \( A + B > C \) and \( B + C > A \), it can form an isosceles triangle. Resultant can be zero.
- **Option D**: If \( A + B < C \), this violates the triangle inequality. Therefore, they cannot form a closed triangle and cannot sum to zero.

4. **Conclusion**: The set of vectors in **Option D** cannot have a resultant of zero because the triangle inequality is not satisfied.

### Final Answer:
The resultant cannot be zero for the set of vectors in **Option D**.