Why the Cross Product of Two Coplanar Vectors is Not Always Zero: Exploring the Common Misunderstanding

Often in mathematical discussions, the statement that the cross product of two coplanar vectors is zero is debated. This article aims to clarify this common misconception by explaining the conditions under which the cross product is zero and why it is not always the case for coplanar vectors. We will also discuss the related concepts, such as vector and scalar triple products, to gain a deeper understanding of vector algebra.

Introduction to Vectors and the Cross Product

In mathematics and physics, vectors are used to represent quantities that have both magnitude and direction. One of the key operations involving vectors is the cross product, denoted as vec{a} times vec{b}, which produces a vector that is perpendicular to both vec{a} and vec{b} in the three-dimensional space. The magnitude of this resulting vector is given by the formula:

|vec{a} times vec{b}| | vec{a} | times; | vec{b} | times; sinθ,

where θ is the angle between vec{a} and vec{b}.

Why the Cross Product Vanishes for Parallel Vectors

It is a well-known fact that the cross product of two parallel vectors (or collinear vectors) is zero. This can be seen from the definition of the cross product, where the sine of the angle between the vectors plays a critical role:

vec{a} times vec{b} | vec{a} | times; | vec{b} | times; sinθ

When the vectors are parallel, the angle θ between them is either 0° or 180°. Since sin(0°) 0 and sin(180°) 0, in both cases, the result of the cross product is zero.

Physically, this makes sense: if two vectors are parallel, the area spanned by a parallelogram formed by these vectors is zero, leading to a zero cross product.

Coplanar Vectors and the Cross Product

Two vectors are always coplanar, meaning they lie in the same plane. However, the cross product of two coplanar vectors is not necessarily zero. The confusion primarily arises because the term "coplanar" is generalized to include both parallel and non-parallel vectors. Here's a clearer way to understand this:

The cross product is zero if and only if the two vectors are parallel or antiparallel. If the vectors are not parallel, the cross product can have a non-zero magnitude, as it represents the area of the parallelogram spanned by the vectors.

For example, consider the unit vectors i and j which lie in the xy-plane. While both i and j are coplanar, their cross product is:

i times j k,

demonstrating that the cross product can be non-zero even for coplanar vectors.

Scalar Triple Product and Vector Triple Product

The distinction between the cross product and the scalar triple product is important in understanding the behavior of vectors in three-dimensional space. The scalar triple product of three vectors vec{a}, vec{b}, and vec{c} is given by the dot product of vec{a} with the cross product of vec{b} and vec{c}:

vec{a} middot; (vec{b} times vec{c}).

If the three vectors are coplanar, the scalar triple product is zero. This is because the cross product vec{b} times vec{c} of two co-planar vectors is zero (as explained earlier), and the dot product of any vector with zero is zero. The vector triple product involves a similar operation but results in a vector, not a scalar, and its behavior in the context of coplanar vectors is different.

In conclusion, while the cross product of two parallel vectors is always zero, the cross product of two coplanar vectors that are not parallel is not zero. Understanding the nuances of these concepts is crucial for advanced applications in physics, engineering, and mathematics. Whether you are dealing with parallel vectors or coplanar vectors, the cross product's value depends on the angle between them, which brings a rich richness to the field of vector algebra.