Yes, it is possible for three vectors of different magnitudes to add to zero.
Understanding Vector Addition
Vector addition is different from adding scalar quantities (like speed or mass). Vectors have both magnitude (size) and direction. When adding vectors, you must account for both. The sum of vectors is often called the resultant vector.
How Unequal Vectors Can Sum to Zero
For a set of vectors to add up to the zero vector (a vector with zero magnitude and no specific direction), when placed head-to-tail, they must form a closed loop, returning to the starting point.
As stated in the reference:
"Yes, three vectors having unequal magnitude can add up to give a zero vector. This can occur in the case of a triangle whose each vertex is formed from a head and tail of successive vectors."
This "case of a triangle" is the key. If you have three vectors, A, B, and C, their sum is zero (A + B + C = 0) if, when you place the tail of B at the head of A, and the tail of C at the head of B, the head of C lands exactly on the tail of A. This configuration forms a triangle.
- Vector A + Vector B = Resultant Vector R₁
- For the sum to be zero, R₁ + Vector C = 0.
- This means Vector C must be equal in magnitude and opposite in direction to Resultant Vector R₁ (i.e., C = -R₁).
Geometrically, if A and B form two sides of a triangle, their resultant R₁ is the third side, pointing from the tail of A to the head of B. For A + B + C = 0, C must close the triangle by connecting the head of B back to the tail of A.
Conditions for Forming a Triangle
For three line segments (which represent vector magnitudes) to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. This is known as the Triangle Inequality Theorem.
So, for three vectors with different magnitudes (|A|, |B|, |C|) to add to zero, they must be able to form a triangle. This means:
- |A| + |B| > |C|
- |A| + |C| > |B|
- |B| + |C| > |A|
As long as these conditions are met, three vectors with unequal magnitudes can be oriented in space (given appropriate directions) such that their vector sum is zero. They cannot be collinear (all lying on the same line) if their magnitudes are non-zero and unequal, as collinear vectors adding to zero generally requires specific magnitude relationships (e.g., two opposing one, or pairs cancelling) that are very restrictive for unequal magnitudes. The ability to form a triangle allows for non-collinear arrangements where the directions balance out the different magnitudes.
Example Scenario
Imagine forces acting on an object. If three forces of different magnitudes (e.g., 5 N, 7 N, and 10 N) are applied to an object, and these forces are directed such that they form a closed triangle when placed head-to-tail, the net force on the object will be zero, and the object will remain in equilibrium (or continue in its state of motion).
