Vector division is not a standard or well-defined operation because there is no universally accepted method to divide one vector by another in a way that satisfies the properties expected of division (like yielding the original vector when multiplied back).
Unlike scalar numbers, vectors possess both magnitude and direction, existing in multi-dimensional spaces. Standard arithmetic operations like addition and subtraction translate readily to vectors, and multiplication has defined forms (dot product and cross product), but division does not have a direct, analogous operation that works universally for all vectors.
One fundamental reason is Dimensional Inconsistency. As highlighted by the reference, vectors can exist in different dimensions (e.g., a 2D vector in a plane or a 3D vector in space). A core requirement for division, even in scalar arithmetic, is that the quantities involved often need to have compatible units or dimensions for the result to be meaningful. With vectors, attempting to divide a 3D vector by a 2D vector, for instance, immediately presents a dimensional mismatch that has no clear interpretation. Even dividing two vectors of the same dimension doesn't resolve the issue because the nature of vector multiplication doesn't produce results (scalars or specific types of vectors) that allow for a simple inverse division operation.
Here's a breakdown of the key reasons why vector division lacks definition:
- Dimensional Inconsistency: Vectors exist in spaces of varying dimensions (2D, 3D, etc.). Division, as typically understood, requires operands to have consistent dimensions or units. Dividing a vector of one dimension by a vector of another dimension is mathematically undefined in standard vector algebra. Even within the same dimension, the geometric and algebraic properties do not lend themselves to a division operation.
- Lack of a Unique Inverse Operation: Division is essentially the inverse of multiplication. For scalars, dividing by
ais the inverse of multiplying bya. However, vector multiplication takes different forms (dot product yields a scalar, cross product yields a vector perpendicular to the plane of the inputs, defined only in 3D).- The dot product results in a scalar. You cannot "divide" a vector by a scalar result to uniquely recover the original vector.
- The cross product results in a vector. However, this operation is not invertible in the way required for division. Knowing the cross product of two vectors doesn't uniquely tell you the original vectors.
- Nature of Vector Spaces: Vector spaces are designed around operations like addition, scalar multiplication, dot products, and cross products (in certain dimensions). Division doesn't fit naturally into the algebraic structure and properties (like associativity, distributivity, etc.) that define vector spaces.
Comparison: Scalar vs. Vector Operations
| Operation | Scalars (Numbers) | Vectors |
|---|---|---|
| Addition | $a+b$ | $\mathbf{v}_1 + \mathbf{v}_2$ |
| Subtraction | $a-b$ | $\mathbf{v}_1 - \mathbf{v}_2$ |
| Multiplication | $a \times b$ (or $a \cdot b$) | Scalar $\times$ Vector ($c \mathbf{v}$) |
| Dot Product ($\mathbf{v}_1 \cdot \mathbf{v}_2$, results in a scalar) | ||
| Cross Product ($\mathbf{v}_1 \times \mathbf{v}_2$, results in a vector, only in 3D) | ||
| Division | $a \div b$ (or $\frac{a}{b}$) | Not defined in standard vector algebra |
While there's no general vector division, certain advanced mathematical fields explore concepts that bear some resemblance to division for specific algebraic structures (like quaternion division in 4D), but these are not part of standard vector algebra.
In summary, the fundamental reasons vector division is not possible in standard vector algebra are the inherent dimensional inconsistencies between vectors and the lack of an inverse operation that behaves like division for the defined types of vector multiplication (dot and cross products).
