Geometric interpretation of Given Vector Functions

Question

Problem : Give a Geometric Interpretation of

Here are different parts of problem and my attempts. I don’t know if it is enough of an interpretation or if there is another type of answer being asked for altogether.

• $ f\begin{pmatrix}\begin{pmatrix} x_1 \\x_2 \\x_3 \\\end{pmatrix}\end{pmatrix}=\begin{pmatrix} -x_1 \\x_2 \\-x_3 \\\end{pmatrix}$
  
  My Attempt : $\mathbb{R^3}\to\mathbb{R^3}$ reflected along the $y-axis$.
  
  
  

• $ f\begin{pmatrix}\begin{pmatrix} x_1 \\x_2 \\x_3 \\\end{pmatrix}\end{pmatrix}=\begin{pmatrix} 0 \\x_2 \\x_3 \\\end{pmatrix}$
  
  My Attempt : $\mathbb{R^3}\to\text{a plane in }\mathbb{R^3}\text{ where }x_1=0$
  
  
  

• $ f\begin{pmatrix}\begin{pmatrix} x_1 \\x_2 \\\end{pmatrix}\end{pmatrix}=\begin{pmatrix} x_2 \\x_1 \\\end{pmatrix}$
  
  My Attempt : $\mathbb{R^3}\to\text{reflects }\mathbb{R^2}\text{ along }y=x$
  
  
  

Answer 1

The first map is a half-turn around the $y$-axis, that is, a rotation of $\pi$ radians.

The second map is the orthogonal projection from $\mathbb{R}^3$ onto the plane $z=0$, with respect to the usual inner product.