Q1: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $U \subseteq V$ be a finite-dimensional subspace. For a vector $x \in V$ we have the orthogonal projection $x = x|U + x|{U^{\perp}}$. What is the distance between $x$ and $U$?

A1: $| x|_U | $

A2: $| x|_{U^{\perp}}| $

A3: $| x | $

A4: $| x + x|_{U^{\perp}} | $

Q2: Let $V$ be a $\mathbb{F}$-vector space with inner product $\langle \cdot, \cdot \rangle$ and $u,v \in V$ two vectors with $u \perp v$. What is always correct?

A1: $| u + v |^2 = | u |^2 + |v |^2$

A2: $| u + v | = | u | + |v |$

A3: $| u + v | = | u |^2$

A4: $| u + v |^2 = 0$

Q3: Let $\mathbb{R}^3$ given with the standard inner product $\langle \cdot, \cdot \rangle$ and $U$ the two-dimensional subspace given by the $x$-$y$-plane. What is the distance between $U$ and $x= \begin{pmatrix} 5 \ 4 \ 3 \end{pmatrix}$?

A1: 5

A2: 4

A3: 3

A4: 0