1. Plug the Point: For each $(x,y)$, find the vector $\mathbf{F}(x,y) = \langle P, Q \rangle$.
2. Draw the Arrow: The arrow starts at $(x,y)$ and points in the direction of the vector.
3. Scale for Clarity: Draw your arrows small (e.g., 1/4 size) so they don't overlap.
4. Identify the Flow: Is the field radial (pointing in/out) or rotational (spinning)?
Given the Vector Field $\mathbf{F}(x, y) = \langle y, -x \rangle$.
The First Quarter: Calculate the vectors for these points:
$(1,0), (0,1), (-1,0), (0,-1)$.
The Diagonal Check: Calculate the vectors for $(1,1)$ and $(-1,-1)$.
On the grid below, draw the vectors you calculated in Part I.
Use a small scale. What pattern do you see emerging?
Is it a Fountain or a Whirlwind?
What is the value of the vector field $\mathbf{F} = \langle x, y \rangle$ at the point $(0,0)$? What is the length of the arrow? Why is the origin often a place of zero "Force" but maximum "Potential"?
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Identify each field as Radial (R) or Rotational (Rot).
$\\mathbf{F} = \langle 2x, 2y \rangle$
$\\mathbf{F} = \langle -y, x \rangle$
$\\mathbf{F} = \langle \frac{x}{\sqrt{x^2+y^2}}, \frac{y}{\sqrt{x^2+y^2}} \rangle$
(Hint: This is a Unit Radial field).
A potential function is $f(x, y) = x^2 + y^2$.
The vector field is its Gradient: $\mathbf{F} = \nabla f$.
Task: Find the formula for $\mathbf{F}$. Calculate the vector at $(1, 2)$. Is this a "Conservative" field?
Objective: Explain Vector Fields to a younger sibling using two magnets.
The Activity:
1. Put two magnets near each other.
2. Have them feel the push/pull without the magnets touching.
3. Ask: "Is the space between them empty? Or is it full of invisible power?"
The Lesson: "Everywhere we go, God has put 'Invisible Arrows' that help us know which way to move. We just have to listen to the Spirit's push."
Response: ___________________________________________________________