1. **Problem statement:** Given two vectors $\vec{a}, \vec{b} \in \mathbb{R}^2$, determine which statements about vectors and their scalar product are true. 2. **Recall the scalar product definition:** The scalar product (dot product) of two vectors $\vec{a} = (a_1, a_2)$ and $\vec{b} = (b_1, b_2)$ is defined as $$\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2,$$ which is a scalar (a real number), not a vector. 3. **Evaluate each statement:** - "Das Skalarprodukt zweier Vektoren $\vec{a} \cdot \vec{b}$ ist ein Vektor." This is false because the scalar product results in a scalar, not a vector. - "$(\vec{a} + \vec{b}) \cdot 5 \in \mathbb{R}$" Here, $\vec{a} + \vec{b}$ is a vector, and multiplying by 5 scales the vector, so $(\vec{a} + \vec{b}) \cdot 5$ is a vector, not a scalar. The statement claims it is in $\mathbb{R}$ (a scalar), so this is false. - "Ist das Skalarprodukt $\vec{a} \cdot \vec{b} = 0$, so stehen die beiden Vektoren $\vec{a}$ und $\vec{b}$ normal aufeinander." If the scalar product is zero, the vectors are orthogonal (perpendicular). This is true. - "Die Vektoren $\vec{a}$ und $7 \cdot \vec{a}$ sind parallel." Multiplying a vector by a scalar changes its magnitude but not its direction, so these vectors are parallel. This is true. 4. **Summary of correct statements:** - $\vec{a} \cdot \vec{b}$ is not a vector but a scalar. - $(\vec{a} + \vec{b}) \cdot 5$ is a vector, not a scalar. - If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a}$ and $\vec{b}$ are orthogonal. - $\vec{a}$ and $7 \cdot \vec{a}$ are parallel. **Final answer:** The true statements are: - "Ist das Skalarprodukt $\vec{a} \cdot \vec{b} = 0$, so stehen die beiden Vektoren normal aufeinander." - "Die Vektoren $\vec{a}$ und $7 \cdot \vec{a}$ sind parallel."