**-A)**

**-B)**

**-C)**

**-D)**

This problem aims to familiarize us with the concept of a **vector field **and **vector space.** The problem is related to vector **calculus** and **physics**, where we will briefly discuss about **vector** **fields** and **spaces**.

When we talk about **vector** **field** in **vector** **calculus** and **physics**, it is a selection of a vector to every individual point in a **subset** of **space**. For illustration, a vector field in the 2-**dimensional** plane can be envisioned as a cluster of **arrows** with an allocated **numerical** **value** and **direction**, each connected to a point in that plane.

**Vector** **fields** are universal in engineering and sciences, as they represent things like **gravity**, **fluid** **flow** **velocity**, **heat** **diffusion**, etc.

## Expert Answer

A **vector** **field** on an area $D$ of $R^2$ is a function $F$ that gives to each point $(x,y)$ in $D$ a vector $F(x,y)$ in $R^2$; in different terms, two **scalar** **functions** are formed $P(x,y)$ and $Q(x,y)$, forming:

\[F(x,y) = P(x,y)\hat{i} + Q(x,y)\hat{j} = < P(x,y), Q(x,y)>\]

This vector field might look like a function that **inputs** a **position** **vector** $ <x, y> $ and **outputs** a **vector** $ <P, Q> $, which is indeed an alteration from a **subset** of $R^2$ to $R^2$. This implies that the **graph** of this vector field spreads in $4$ **dimensions**, but there is an **alternative** way to graph a **vector** **field**, which we will graph in a minute.

So in order to figure out the **correct** **option** from the given choices, we will take some **random** points and will plot them against the given **equation** that is $F(x,y) = <x, -y>$.

Thus, now taking the **point** $(x,y)$ and **computing** the $F(x,y) = <x, -y>$:

\[(1, 0) = <1, 0>\]

\[ (0, 1) = <0, -1>\]

\[ (-1, 0) = <-1, 0>\]

\[ (0, -1) = <0, 1> \]

\[ (2, 0) = <2, 0> \]

\[ (0, 2) = <0, -2> \]

The **evaluations** of the vector field at the assumed **points** are $ <1, 0>, <0, -1>, <-1, 0>, <0, 1>, <2, 0>, <0, -2> $ respectively. Now **plotting** the vector field of the above points:

Clearly all points from the $1^{st}$ **quadrant** map to all points of the $4^{th}$ **quadrant** and so on. Similarly all points of the $2^{nd}$**quadrant** map to all points of $3^{rd}$ **quadrant** and so on.

## Numerical Answer

Hence, the **answer** is option $D$:

## Example

Plot the **vector** **field** $ F(x,y) = <1, x> $.

We will take the **point** $(x,y)$ and **compute** the $F(x,y) = <1, x>$:

\[ (-2, -1) = <1, -2> \]

\[ (-2, 1) = <1, -2> \]

\[ (-2, 3) = <1, -2> \]

\[ (0, -2) = <1, 0> \]

\[ (0, 0) = <1, 0> \]

\[ (0, 2) = <1, 0> \]

\[ (2, -3) = <1, 2> \]

\[ (2, -1) = <1, 2> \]

\[ (2, 1) = <1, 2> \]

Now **plotting** the **vector** **field** of the above **points**: