Three displacement vectors have lengths 1 m, 2 m, and 4 m. Could they possibly add together to get a vector of length zero?
The Correct Answer and Explanation is :
No, the three displacement vectors of lengths 1 m, 2 m, and 4 m cannot add together to result in a vector of length zero.
Explanation:
To understand why, consider the conditions required for three vectors to sum to zero. The vectors must form a closed triangle because, geometrically, a closed triangle ensures that the sum of the vectors is zero (the head of the last vector meets the tail of the first vector).
Triangle Inequality Theorem:
For three vectors to form a triangle:
- The sum of the lengths of any two sides must be greater than or equal to the length of the third side.
- Mathematically:
- a+b≥ca + b \geq c
- b+c≥ab + c \geq a
- c+a≥bc + a \geq b
For the given vectors of lengths 1 m, 2 m, and 4 m:
- Check 1+2≥41 + 2 \geq 4: False (1 + 2 = 3, which is less than 4).
- Check 2+4≥12 + 4 \geq 1: True (2 + 4 = 6, which is greater than 1).
- Check 4+1≥24 + 1 \geq 2: True (4 + 1 = 5, which is greater than 2).
Since the first condition (1+2≥41 + 2 \geq 4) fails, the three vectors cannot form a triangle. As a result, they cannot sum to zero.
Intuition:
When trying to add these vectors head-to-tail, the longest vector (4 m) cannot be balanced by the combined lengths of the other two vectors (1 m and 2 m). The inability to form a closed loop means the resultant vector cannot be zero.
Thus, it is geometrically and mathematically impossible for these vectors to sum to zero.