(A) (5 marks) Consider the euclidean space $\mathbb { R } ^ { n }$ with the usual norm and dot product. Let $\mathrm { x } , \mathrm { y } \in \mathbb { R } ^ { n }$ be such that $$\| \mathrm { x } + t \mathrm { y } \| \geq \| \mathrm { x } \| , \text { for all } t \in \mathbb { R }$$ Show that $\mathbf { x } \cdot \mathbf { y } = 0$. (B) (5 marks) Consider the vector field $\vec { v } = \left( v _ { x } , v _ { y } \right)$ (with components $\left( v _ { x } , v _ { y } \right)$ ) on $\mathbb { R } ^ { 2 }$: $$v _ { x } ( x , y ) = x - y , v _ { y } ( x , y ) = y + x$$ Compute the line integral of $\vec { v }$ along the unit circle (counterclockwise). Is there a function $f$ such that $\vec { v } = \operatorname { grad } f$?
(A) (5 marks) Consider the euclidean space $\mathbb { R } ^ { n }$ with the usual norm and dot product. Let $\mathrm { x } , \mathrm { y } \in \mathbb { R } ^ { n }$ be such that
$$\| \mathrm { x } + t \mathrm { y } \| \geq \| \mathrm { x } \| , \text { for all } t \in \mathbb { R }$$
Show that $\mathbf { x } \cdot \mathbf { y } = 0$.\\
(B) (5 marks) Consider the vector field $\vec { v } = \left( v _ { x } , v _ { y } \right)$ (with components $\left( v _ { x } , v _ { y } \right)$ ) on $\mathbb { R } ^ { 2 }$:
$$v _ { x } ( x , y ) = x - y , v _ { y } ( x , y ) = y + x$$
Compute the line integral of $\vec { v }$ along the unit circle (counterclockwise). Is there a function $f$ such that $\vec { v } = \operatorname { grad } f$?