In case of three vector quanitites of same type, whose resultant cannot be zero ?

Options

A. 120, 10, 10
B. 10, 10, 20
C. 10, 20, 20
D. 10, 20, 40

Explanation

To determine the three vector quantities of the same type whose resultant cannot be zero, we need to apply the triangle inequality theorem. According to this theorem, for three vectors to form a closed triangle (resultant being zero), the sum of the lengths of any two sides must be greater than the length of the third side. If this condition is not satisfied, the resultant cannot be zero.

### Step-by-Step Solution:

1. **Understanding the Condition**:
- For three vectors \( \vec{A}, \vec{B}, \vec{C} \) to have a resultant of zero, they must satisfy the triangle inequality:
- \( |\vec{A}| + |\vec{B}| > |\vec{C}| \)
- \( |\vec{A}| + |\vec{C}| > |\vec{B}| \)
- \( |\vec{B}| + |\vec{C}| > |\vec{A}| \)

2. **Identifying the Non-Zero Resultant Condition**:
- If the sum of the lengths of any two vectors is less than or equal to the length of the third vector, the resultant cannot be zero. This can be expressed as:
- \( |\vec{A}| + |\vec{B}| \leq |\vec{C}| \)
- \( |\vec{A}| + |\vec{C}| \leq |\vec{B}| \)
- \( |\vec{B}| + |\vec{C}| \leq |\vec{A}| \)

3. **Evaluating Given Options**:
- Let's evaluate the options provided:
1. **Option 1**: \( 120, 10, 10 \)
- \( 120 > 10 + 10 \) (True)
- Resultant cannot be zero.
2. **Option 2**: \( 10, 10, 20 \)
- \( 10 + 10 = 20 \) (Not less than)
- Resultant can be zero.
3. **Option 3**: \( 10, 20, 20 \)
- \( 10 + 20 > 20 \) (True)
- Resultant can be zero.
4. **Option 4**: \( 10, 20, 40 \)
- \( 10 + 20 = 30 < 40 \) (True)
- Resultant cannot be zero.

4. **Conclusion**:
- The vectors \( 10, 20, 40 \) satisfy the condition where the sum of two sides is less than the third side. Therefore, the resultant of these vectors cannot be zero.

### Final Answer:
The three vector quantities whose resultant cannot be zero are \( 10, 20, 40 \).