Can Vectors of Unequal Magnitudes Add Up to Zero?

In the realm of vector spaces, the concept of vectors adding up to the zero vector is an intriguing one. Given two vectors of unequal magnitudes, can they sum to zero? This article delves into the conditions under which this scenario is possible and provides insights through mathematical proofs and examples.

Understanding Vector Addition in a Vector Space

In a vector space, if you have two vectors a and b, their sum a b can only be the zero vector 0 under specific conditions. This is fundamentally related to the properties of additive inverses in vector spaces.

For any vector v in a vector space, there exists a unique inverse vector -v, such that:

[ v (-v) 0 ]

This inverse vector -v is the only solution to the equation av 0, whereas vw 0 holds true if and only if w -v.

Examples with Unequal Magnitudes

Consider the example where two numbers are unequal but add up to zero. For instance, 5 and -5. Similarly, vectors can have different magnitudes but still sum to zero under certain conditions.

Let each of the vectors a and b be members of a normed linear space. Suppose:

[ a b 0 ]

It implies that:

[ a -b ]

Therefore, the norms (magnitudes) of a and b are equal:

[ |a| |b| ]

As a result, if the sum of two vectors in a normed linear space is zero, then the vectors must have equal magnitudes.

Complex Scenarios with Multiple Vectors

The above condition is true for two vectors. However, if you consider more than two vectors in a normed linear space, their sum can be zero even if their norms are not equal. For example, if you have three non-zero vectors:

a, b, and c, their sum can still be zero, which is only possible if their norms are not all equal.

A concrete example is a set of three complex vectors with equal norms:

[ -4 3i, 1 - 2i, 3 - i, 4 2i, 1 - 2i, -5 0 ]

Each of these vectors has the same norm:

[ sqrt{30} ]

For instance, if you sum the vectors:

[ (-4 3i) (1 - 2i) (3 - i) (4 2i) (1 - 2i) (-5) (0 0) (0 0)i 0 0i 0 ]

This demonstrates that even with unequal magnitudes, the vectors can still sum to zero in specific contexts.

Contradictory Assumptions

Consider the assumption that two vectors v and w of unequal magnitudes can sum to the zero vector:

[ v w 0, text{ where } w eq v ]

If neither v nor w is the zero vector, they must be collinear (i.e., they lie on the same line).

Let i be the unit vector in the direction of v. Then:

[ v xi, , w yi ]

Since i is a unit vector:

[ v x, , w y ]

The equation xi yi 0 leads to:

[ xyi 0 Rightarrow xy 0 Rightarrow x -y ]

If x ! y, then x ! -y, leading to a contradiction. Thus, the two vectors cannot have unequal magnitudes and still sum to zero.

In summary, the only way two vectors can combine to the zero vector are:

Both vectors are the zero vector. The vectors have equal magnitudes and opposite directions.

In both cases, the two vectors have equal magnitudes, hence they cannot have unequal magnitudes.