What is the minimum number of non zero, non collinear vectors required so that their vector sum is the zero vector?

Introduction / Context:
Vector addition is a fundamental topic in physics and mathematics. When several forces or displacements act together, we often represent them as vectors and add them using the triangle or polygon law. This question asks for the smallest number of non zero, non collinear vectors that can add up to the zero vector. It checks your understanding of both the geometry and the algebra of vector addition.

Given Data / Assumptions:

• All vectors considered are non zero, so none of them has zero magnitude.
• The vectors are non collinear, so no two of them lie exactly along the same straight line in opposite directions.
• The vector sum of all the chosen vectors is required to be the zero vector (resultant equal to zero).
• We can arrange the vectors head to tail according to the polygon law of vector addition.

Concept / Approach:
The sum of vectors is represented geometrically by joining them head to tail. If the final head coincides with the initial tail, the resultant is the zero vector and the vectors form a closed polygon. With two vectors, you can get a zero resultant only if they are equal in magnitude and opposite in direction, which makes them collinear. That violates the non collinear condition. With three vectors, you can arrange them tip to tail to form a closed triangle, with each vector pointing in a different direction, and still satisfy the zero resultant condition. Therefore, three is the minimum number.

Step-by-Step Solution:
Step 1: Consider one non zero vector. It obviously cannot add to zero by itself, because it has some magnitude and direction. Step 2: Consider two non zero vectors. For their sum to be zero, the second must be equal in magnitude and opposite in direction to the first. Step 3: Equal and opposite vectors lie along the same line, which means they are collinear and violate the non collinear condition in the problem. Step 4: Now consider three vectors. If we can arrange them in such a way that they form a closed triangle when placed head to tail, then the net displacement is zero. Step 5: In a closed triangle, none of the sides is necessarily collinear with the others; all three can have different directions while still adding to zero. Step 6: Hence, three non zero, non collinear vectors can have a zero vector sum, and fewer than three cannot satisfy all conditions.

Verification / Alternative check:
Algebraically, if two vectors A and B are non zero and non collinear, then A + B cannot be zero because that would imply B = minus A, which makes them collinear and opposite. For three vectors A, B and C, the condition A + B + C = 0 can be written as A + B = minus C, which is geometrically the triangle law of addition. Many physical examples, such as three forces in equilibrium on a rigid body, show that three forces of appropriate magnitudes and directions can balance each other without being collinear, forming a closed triangle in the vector diagram. This confirms that three is sufficient and minimum.

Why Other Options Are Wrong:
2: Two vectors can sum to zero only if they are equal and opposite, which makes them collinear, violating the non collinear condition.
4: Four vectors can certainly sum to zero by forming a closed quadrilateral, but three vectors already satisfy the conditions, so four is not the minimum number.
1: A single non zero vector can never have a zero resultant, because there is no other vector to cancel it.

Common Pitfalls:
Some learners quickly choose two vectors because they remember that a pair of equal and opposite vectors can cancel out. They forget the additional requirement that vectors must be non collinear. Others may think that a more complex shape than a triangle is needed and choose four. To avoid such errors, always check all the given conditions and remember the geometric interpretation of vector addition as a closed polygon. The triangle is the simplest closed polygon, so three is the correct minimum.

Final Answer:
The minimum number of non zero, non collinear vectors required to produce a zero vector is 3.