Units

# Unit 11 : Vector Fields in 2D and 3D

## The Concept

A **vector field** is an assignment of a vector to each point in a subset of space. In other words, if we are given a vector [latex]\langle x,y\rangle[/latex], then the vector is simply the mapping in 2D of each point.

Vector fields can be written in two equivalent notations shown below for both 2D and 3D:

**2D Notation:**[latex]\vec{F}(x, y) = P(x, y)\vec{i} + Q(x, y)\vec{j} = \langle P(x, y) , Q(x, y)\rangle[/latex]**3D Notation:**[latex]\vec{F}(x,y,z) = P(x, y,z)\vec{i}+ Q(x,y,z)\vec{j} + R(x,y,z)\vec{k} = \langle P(x, y,z) , Q(x, y,z) , R(x,y,z)\rangle[/latex]

Where [latex]\vec{i} = \langle 1 , 0 \rangle[/latex] and [latex]\vec{j} = \langle 0,1\rangle[/latex] represent **unit vectors** in 2D, and [latex]\vec{i} = \langle1,0,0\rangle ,\, \vec{j} = \langle 0,1,0 \rangle[/latex] and [latex]\vec{k} = \langle 0,0,1\rangle[/latex] are unit vectors in 3D. A real life example that can be modeled as a vector field would be a fluid dynamics problem such as a river, where the velocity of the liquid is a vector at any given point. The magnitude (i.e., amplitude) of the vector represents the speed and the direction represents the direction of the flow at any given point.

## The Plot

Now, you should engage with the 2D and 3D plots below to understand 2D and 3D vector fields^{[1]}. Follow the steps below to apply changes to the plot and observe the effects:

- The vector definition is done using [latex]P[/latex] and [latex]Q[/latex].
- Y Grid and X Grid control the number of arrows that will appear in the 2D plot.
- Xmin and Ymin set the minimum boundaries for the plot.
- YMax and Xmax set the maximum boundaries for the plot.

### 2D Vector Field Plot

### 3D Vector Field Plot

## Self-Checking Questions

Check your understanding by solving the following questions^{[2]}:

- Draw the following vector field [latex]\vec{F}(x,y)=x\vec{i} + y\vec{j}[/latex]
- Draw the following vector field [latex]\vec{F}(x, y, z) = 2x\vec{i} − 2y\vec{j} − 2z\vec{k}[/latex]

Use the graphs to find the answers to these questions.