Lab 09: Length and Distance

Interactive tools for Chapter 9. Move vectors, compare data points, change feature scales, explore nearest neighbors, and watch distances behave strangely in high dimensions.

1. Vector Length

x-coordinate: y-coordinate:

Length is the size of a vector. In two dimensions, it is the diagonal length from the Pythagorean theorem.

2. Distance Between Two Points

x₁: x₂: y₁: y₂:

3. Distance Rules: L₁, L₂, and L∞

difference a: difference b:

Different distance rules answer different questions. L₁ counts coordinate movement, L₂ measures straight-line distance, and L∞ asks for the worst coordinate difference.

4. Feature Scaling Trap

We compare a query apartment to three apartments using rent, size, distance-to-campus, and bedrooms. Use the slider to change the weight of rent. Notice how the nearest neighbor can change.

rent scale multiplier:

Raw distance can be dominated by the largest numerical feature. A distance calculation is also a modeling decision.

5. Nearest Neighbor Classifier

new point x: new point y:

6. Image Distance

A small image can be flattened into a vector. Pixel distance is easy to compute, but it may not always agree with human visual similarity.

shift image B horizontally: brightness of image C:

7. High-Dimensional Distance

number of random pairs:

As dimension grows, distances usually grow. Their relative spread often shrinks. This is one reason high-dimensional geometry can feel unlike the plane.

8. Matrix Transformations Change Distance

horizontal stretch: vertical stretch: rotation angle:

Reflection Questions

What is the difference between a vector and its length?
Why is distance between data points not automatically meaningful?
When might Manhattan distance be better than Euclidean distance?
What does standardization do before a distance calculation?
Why can high-dimensional distance be surprising?