as plt import numpy as np

Define the points P and Q

P = np.array([2, 1, 0]) Q = np.array([-1, 2, -4])

Plot the points

fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.scatter(*P, color='blue', label='P(2, 1, 0)', s=50) ax.scatter(*Q, color='red', label='Q(-1, 2, -4)', s=50)

Draw a line between P and Q for visual clarity

ax.plot([P[0], Q[0]], [P[1], Q[1]], [P[2], Q[2]], color='black', linestyle='--')

Label axes

ax.set_xlabel('X-axis') ax.set_ylabel('Y-axis') ax.set_zlabel('Z-axis')

#(a) The points $P(2, 1, 0)$ and $Q(-1, 2, -4)$ have been plotted in 3D space.

(b) The distance between $P$ and $Q$ is approximately $5.10$ units, calculated using the Euclidean distance formula:

$d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$

Would you like a detailed step-by-step explanation of the calculation or anything else clarified?

Here are five follow-up questions related to this:

1. How is the distance formula derived in 3D space?
2. Can you find the midpoint of the line segment connecting $P$ and $Q$?
3. How would you calculate the vector from $P$ to $Q$?
4. Can you describe the direction of the line passing through $P$ and $Q$?
5. What is the significance of the distance in geometric applications?

Tip: Visualizing points and their connections in 3D space helps in understanding spatial relationships and vector properties.