describe a Function geometrically

Question

Question is :

Let $v ∈ \mathbb{R^n}$ be a vector with $∥v∥ = 1$.

Consider the mapping $φ_v: \mathbb{R^n} → \mathbb{R^n}$ given by :

$ φ_v (x) = x - 2⟨x, v⟩v.$

Describe the function geometrically.

I'm not sure that i'm understanding the question correctly , how can i describe a mapping geometrically ?

Answer 1

The question is asking what do you geometrically do to $\vec{x}$ to get $\phi_v\left(\vec{x}\right)$.

For example, if $f\left(\vec{x}\right) = \vec{x} + \vec{1}$, then $f$ geometrically is a translation of $\vec{x}$ by the vector of all ones.

To describe your problem, think of how you compute projections as a hint.