Understanding Vector Spaces: Why 0 1 and 1 0 Are Perpendicular
The concept of vector spaces can be initially perplexing because vectors are often presented as simple lists of numbers or tuples. However, visualizing vectors in space allows us to understand why certain vectors are perpendicular to each other, despite the inherent limitations of working with lists alone.
Vector Spaces as Sets of Tuples
When we define a vector space, we consider it as a set of tuples, but this doesn't inherently give us the concept of angles and lengths. In a vector space, elements are organized as tuples consisting of numbers, but there is no a priori concept of angles or lengths unless we define additional structure on these vectors. For this structure, we introduce the notion of an inner product space, which endows the vector space with a notion of length and angle.
Inner Product Spaces and the Dot Product
The simplest and most familiar inner product space is the real vector space equipped with the dot product. The dot product is a way to define an inner product on real or complex coordinate vectors in R^n or C^n, respectively. In this context, the dot product of two vectors, say vec{a} (a_1, a_2, ldots, a_n) and vec{b} (b_1, b_2, ldots, b_n), is defined as:
vec{a} cdot vec{b} sum_{i1}^{n} a_i b_i
This formula provides a means to calculate the angle between vectors and their lengths. If the dot product of two vectors is zero, then the vectors are said to be orthogonal or perpendicular.
Visualizing Vectors as Arrows in Space
To better understand why the vectors 0 1 and 1 0 are perpendicular, we need to think of vectors as arrows in space rather than just lists of numbers. In a Euclidean space, represented by a real vector space, vectors can be visualized as arrows with a tail and a tip.
In Euclidean geometry, a vector vec{v} vec{pq} is an arrow with its tail at point text{p} and its tip at point text{q}. The length of the vector is the distance from text{p} to text{q}. If we have two vectors, say vec{v} vec{pq} and vec{w} vec{rs}, and we choose a common point text{x} such that text{p} text{x} text{r}, we can add these vectors by placing the tail of text{w} at the tip of text{v}, resulting in a new vector vec{v} vec{w} vec{ps}.
The vectors 0 1 and 1 0, when visualized as arrows in space, have their tails at the origin (0, 0) and their tips at (0, 1) and (1, 0), respectively. These vectors form a right angle between them. The dot product of these vectors is:
(0, 1) cdot (1, 0) 0 cdot 1 1 cdot 0 0
Since the dot product is zero, these vectors are perpendicular.
Free Vectors and Parallelograms
For a deeper understanding, consider the concept of free vectors. A free vector vec{v} vec{pq} is a vector that can be moved around in a space without changing its magnitude or direction, as long as the distance and orientation remain the same. The parallelogram law, which asserts that the sum of two vectors is the diagonal of a parallelogram formed by the two vectors as adjacent sides, is a fundamental concept in vector addition.
Take the vectors 0 1 and 1 0, both having a length of 1. When added head-to-tail, they form a right angle, indicating that they are indeed perpendicular. This can be visualized in a two-dimensional Euclidean space, where these vectors span an ordinary Euclidean plane.
Summary and Conclusion
In summary, while the definition of a vector space as a set of tuples does not inherently include the concept of angles or lengths, the introduction of an inner product, such as the dot product, allows us to define these concepts. The vectors 0 1 and 1 0 are perpendicular because their dot product is zero, reflecting the right angle between them when visualized as arrows in space.
Understanding the geometric interpretation of vectors is essential for grasping more advanced concepts in linear algebra and vector calculus. By visualizing vectors as arrows and utilizing the dot product, we can effectively determine the relationships between different vectors in a vector space.