Question

The magnitudes of a set of 3 vectors are given below. The set of vectors for which the resultant cannot be zero is

• A. 15, 20, 30
• B. 20, 20, 30
• C. 25, 20, 35
• D. 10, 10, 20
• E. 10, 20, 40

Hint:

For zero resultant of 3 vectors, always check triangle inequality.

Answer 1

Answer: (E)

Concept: Three vectors can have zero resultant only if they can form a closed triangle. This is possible only when the sum of any two sides is greater than or equal to the third side (triangle inequality condition).

Step 1: Write triangle condition. \[ a + b \geq c,\quad b + c \geq a,\quad c + a \geq b \]

Step 2: Check option (A): 15, 20, 30 \[ 15 + 20 = 35 > 30 \quad \text{(valid)} \]

Step 3: Check option (B): 20, 20, 30 \[ 20 + 20 = 40 > 30 \quad \text{(valid)} \]

Step 4: Check option (C): 25, 20, 35 \[ 25 + 20 = 45 > 35 \quad \text{(valid)} \]

Step 5: Check option (D): 10, 10, 20 \[ 10 + 10 = 20 \Rightarrow \text{degenerate case (resultant can be zero)} \]

Step 6: Check option (E): 10, 20, 40 \[ 10 + 20 = 30 < 40 \quad \text{(violates condition)} \]

Step 7: Conclusion. Option (E) cannot form a triangle, hence resultant cannot be zero.