Final answer:
Planar vector fields form a vector space because they satisfy key vector space properties such as vector addition, scalar multiplication, and the existence of a zero vector, where each vector addition or scalar multiplication of fields results in another planar vector field.
Step-by-step explanation:
A planar vector field is a function that assigns a vector to each point in the plane, creating a collection of vectors defined over a region in ℝ². For the set of all planar vector fields to form a vector space, they must satisfy certain properties such as vector addition, scalar multiplication, and the existence of a zero vector.
The vector field satisfies the vector addition property since the sum of any two vector fields is also a vector field. If v(x, y) and w(x, y) are two vector fields, their sum v + w is defined component-wise: (v₁(x, y) + w₁(x, y), v₂(x, y) + w₂(x, y)). The resulting vector at each point is simply the vector addition of v and w at that point, which remains a vector in ℝ².
Scalar multiplication is equally valid in this vector space. If c is any scalar and v(x, y) is a vector field, then the product, cv, is defined by multiplying each component of v by c: (cv₁(x, y), cv₂(x, y)). The resulting field is still a planar vector field.
Further, the zero vector in this context is a field where each vector in the field is the zero vector (0, 0), meeting the requirement for the existence of a zero vector in a vector space.
These examples indicate that planar vector fields adhere to the principles of vector addition, scalar multiplication, and contain a zero vector, among other properties, satisfying the definition of a vector space.