Introduction

The concept of vector addition is fundamental in mathematics and physics. When considering the sum of two vectors, there are several ways to interpret and compute it, depending on the vector space and the context. This article aims to delve into the details of when the sum of two vectors, a and b, equals vector c, exploring the geometric and algebraic interpretations.

Vector Addition in a Cartesian Plane

In a two-dimensional Cartesian plane (Oxy), the sum of two vectors a and b can be understood through the parallelogram rule. If a and b are the directed line segments representing vectors, and they are two adjacent sides of a parallelogram OACB with O as the origin, then the free vector a can be represented by the directed segment OA on the x-axis, while b OB is the other side. The diagonal of the parallelogram OC represents the vector c, which is the sum of vectors a and b.

Mathematically, if vector a is given by a xai yaj and vector b is given by b xbi ybj, then the sum c is given by:

c a b (xa xb)i (ya yb)j

This rule also applies in three-dimensional space (Oijk) with the tri-orthonormal system of coordinates. If c is a given vector, its component vectors a and b with a b c can be found, but additional information on c's position and conditions on a and b are necessary.

Vector Addition in Vector Space

In a more general vector space like R2, the vectors a and b can be expressed as a [xa, ya]T and b [xb, yb]T. The sum c is then given by:

c a b [xa xb, ya yb]T

This result is essentially equivalent to the parallelogram rule due to the isomorphism between the plane Oxy and the vector space R2.

Solving Vector Equations

If c is a given vector, such as c [xc, yc]T, the vector equation system [xa xb, ya yb]T [xc, yc]T represents a 2-by-2 non-homogeneous linear system with four unknowns (xa, ya, xb, yb). Such a system typically has infinitely many solutions, indicating that multiple combinations of vectors a and b can sum to the same vector c.

Conclusion

Understanding the sum of two vectors and when it equals a third vector involves the application of both geometric and algebraic principles. Whether through the parallelogram rule in a two-dimensional plane or the algebraic addition in vector space, the process remains fundamentally consistent. However, the existence of infinitely many solutions highlights the importance of additional conditions in practical applications.