Q1: Let $\mathbf{v},\mathbf{w} \in \mathbb{R}^3$ be given by $\mathbf{v} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix}$ and $\mathbf{w} = \begin{pmatrix} -2 \ 1 \ 0 \end{pmatrix}$. What is the cross product $\mathbf{v} \times \mathbf{w}$?

A1: $0$

A2: $\begin{pmatrix} -1 \ 1 \ 0 \end{pmatrix}$

A3: $\begin{pmatrix} -3 \ 6 \ 6 \end{pmatrix}$

A4: $\begin{pmatrix} -3 \ -6 \ 5 \end{pmatrix}$

Q2: Let $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3 \in \mathbb{R}^3$ be the canonical unit vectors. What is not correct?

A1: $ \mathbf{e}_1 \times \mathbf{e}_3 = \mathbf{e}_2$

A2: $ \mathbf{e}_1 \times \mathbf{e}_2 = \mathbf{e}_3$

A3: $ \mathbf{e}_2 \times \mathbf{e}_3 = \mathbf{e}_1$

A4: $ \mathbf{e}_3 \times \mathbf{e}_1 = \mathbf{e}_2$

A5: $ \mathbf{e}_3 \times \mathbf{e}_2 = -\mathbf{e}_1$

Q3: What is not a property of the cross product $ \times: \mathbb{R}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^3$?

A1: $\mathbf{u} \times \mathbf{v} = \mathbf{v} \times \mathbf{u}$ for all $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$.

A2: $\mathbf{u} \times \mathbf{v} = - \mathbf{v} \times \mathbf{u}$ for all $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$.

A3: $\mathbf{u} \times \mathbf{u} = \mathbf{0}$ for all $\mathbf{u} \in \mathbb{R}^3$.

A4: $\mathbf{u} \times \mathbf{0} = \mathbf{0}$ for all $\mathbf{u} \in \mathbb{R}^3$.