Suppose k ≥3, x, y ∈R^k,|x-y|=d>0, and r>0 . Prove: (a) If 2 r>d, there are infinitely many z ∈R

step 1 So we want to prove that if the absolute or the magnitude of z1 plus z2 equals the magnitude of

Suppose k ≥3, x, y ∈R^k,|x-y|=d>0, and r>0 . Prove: (a) If 2...

Suppose $k \geq 3, x, y \in R^{k},|x-y|=d>0$, and $r>0 .$ Prove: (a) If $2 r>d$, there are infinitely many $z \in R^{k}$ such that $$|\mathbf{z}-\mathbf{x}|=|\mathbf{z}-\mathbf{y}|=r$$ (b) If $2 r=d$, there is exactly one such $\mathrm{z}$. (c) If $2 r<d$, there is no such $\mathbf{z}$. How must these statements be modified if $k$ is 2 or $1 ?$

Suppose $k \geq 3, x, y \in R^{k},|x-y|=d>0$, and $r>0 .$ Prove:
(a) If $2 r>d$, there are infinitely many $z \in R^{k}$ such that
$$|\mathbf{z}-\mathbf{x}|=|\mathbf{z}-\mathbf{y}|=r$$
(b) If $2 r=d$, there is exactly one such $\mathrm{z}$.
(c) If $2 r<d$, there is no such $\mathbf{z}$. How must these statements be modified if $k$ is 2 or $1 ?$

Key Concepts

Sphere in Euclidean Space

A sphere in an n-dimensional Euclidean space is the set of all points at a fixed distance (called the radius) from a specific point (called the center). Understanding spheres is central when analyzing problems involving points equidistant from given locations, as the conditions for points lying on spheres lead to concrete algebraic and geometric criteria.

Intersection of Spheres

The intersection of two spheres in Euclidean space is determined by the distance between their centers and their radii. When two spheres share the same radius, points that are equidistant from both centers lie on the intersection which, depending on the relationship between the radius and the separation of centers, can be a sphere of lower dimension (e.g., a circle when k = 3), a single point (if the spheres are tangent), or empty (if the spheres do not meet).

Distance Criteria for Intersection

A key concept is the role of the distance between the centers relative to twice the radius. This condition determines whether the spheres intersect in infinitely many points (when the centers are closer than twice the radius), exactly one point (when the centers are exactly twice the radius apart), or not at all (when the centers are farther apart than twice the radius). This principle underpins the proof structure in problems involving equidistance in Euclidean space.

Dimensional Considerations

The dimensionality of the ambient space significantly affects the nature of the intersection. In higher dimensions (k ? 3), the intersection of spheres can be a manifold (like a circle or sphere of dimension k-2). However, in lower dimensions such as two or one, the intersection changes: in R2, spheres are circles so the intersection can be two points, one point, or none, while in R1, spheres reduce to pairs of points and the conditions for intersection correspondingly simplify. This concept is crucial when modifying proofs or statements for spaces of different dimensions.

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