Concept 02: Dot Products, Projections & Alignment

▶ Interactive Demo: Dot Product & Projection Visualizer

Open the interactive demo below to rotate two vector arrows and see their dot product, angle \θ, and projected shadow update live.

1. The Real-World Problem: Are We Driving on Path?

Imagine your robot is following an autonomous path pointing straight down the field:

• Desired Path Direction: u = [1.0, 0.0] (Straight East).
• Actual Robot Velocity: v = [2.0, 1.5] (Moving at an angle due to wheel slip).

The path-following software needs to know:

1. How much speed is driving along the path? (Forward progress).
2. How much speed is pushing off-course? (Cross-track error).

2. Solving It in Code (Java & WPILib)

First-Principles Java

Calculating dot product, magnitudes, and directional alignment:

// Vector A: Robot Heading Vector (Facing 45 degrees)
double ax = 1.0, ay = 1.0;

// Vector B: Target Line-of-Sight Vector
double bx = 3.0, by = 1.0;

// 1. Compute Dot Product: A · B = ax*bx + ay*by
double dotProduct = ax * bx + ay * by; // 1*3 + 1*1 = 4.0

// 2. Compute Magnitudes
double magA = Math.hypot(ax, ay); // 1.414
double magB = Math.hypot(bx, by); // 3.162

// 3. Cosine of the angle between them: cos(θ) = (A · B) / (|A| * |B|)
double cosTheta = dotProduct / (magA * magB);
double angleDegrees = Math.toDegrees(Math.acos(cosTheta));

System.out.printf("Alignment Angle: %.1f degrees%n", angleDegrees);

Production WPILib Equivalent

import edu.wpi.first.math.geometry.Translation2d;

Translation2d heading = new Translation2d(1.0, 1.0);
Translation2d targetDir = new Translation2d(3.0, 1.0);

// Angle between vectors using WPILib Rotation2d
double angleDiff = heading.getAngle().minus(targetDir.getAngle()).getDegrees();

3. Bridge to Machine Learning: Cosine Similarity

In modern AI search engines and RAG (Retrieval-Augmented Generation):

• When you ask a question to an AI, the question is converted into a vector q.
• Every document in the database is stored as a vector d.
• The AI computes the Cosine Similarity:

  Similarity = (q · d) / (||q|| · ||d||)
  

• The document with the highest dot product is the most relevant answer!

4. Review Checkpoints

Checkpoint 1

Vector u = [3.0, 4.0] and vector v = [-4.0, 3.0]. Compute u · v. What is the angle between them?

Solution:

1. u · v = (3.0)(-4.0) + (4.0)(3.0) = -12.0 + 12.0 = 0.0.
2. Because the dot product is 0.0, the angle between them is exactly 90° (Perpendicular).

Checkpoint 2

A robot applies a force vector F = [10.0, 0.0] Newtons while driving along displacement vector d = [5.0, 2.0] meters. How much mechanical work was done?

Solution: In physics, Work = Force · displacement = (10.0)(5.0) + (0.0)(2.0) = 50.0 Joules. The vertical displacement 2.0m did zero work because it was perpendicular to the force.