Vector Tutorial for Kids and AI Beginners

What is a Vector?

A vector is like a magical arrow that tells you:

• Which direction to go
• How far to go

Example: 3 steps right and 2 steps up → Vector: [3, 2]

Real Life Examples

• GPS: [latitude, longitude]
• Game character movement: [x, y]
• AI: Words, images, and sound are all represented as vectors

Types of Vectors

• 1D: [5]
• 2D: [3, 4]
• 3D: [2, 4, 6]

Basic Vector Operations (Using NumPy)

import numpy as np

1 Create a Vector

v = np.array([3, 4])

2 Add Two Vectors

v1 = np.array([1, 2])
v2 = np.array([3, 4])
print(v1 + v2)
    

3 Subtract Two Vectors

print(v2 - v1)

4 Multiply by Number (Scaling)

print(2 * v1)

5 Dot Product

print(np.dot(v1, v2))

6 Length (Magnitude)

print(np.linalg.norm(v1))

Vector Use in AI

Mini AI Task

goal = np.array([10, 10])
step1 = np.array([8, 9])
step2 = np.array([5, 3])

dist1 = np.linalg.norm(goal - step1)
dist2 = np.linalg.norm(goal - step2)

if dist1 < dist2:
    print("Step1 is closer")
else:
    print("Step2 is closer")
    

Advanced Operations for AI

Normalize a Vector (Unit Vector)

v = np.array([3, 4])
unit_v = v / np.linalg.norm(v)
print(unit_v)
   

Cosine Similarity

def cosine_similarity(a, b):
    return np.dot(a, b) / (np.linalg.norm(a) * np.linalg.norm(b))

v1 = np.array([1, 2])
v2 = np.array([2, 3])
print(cosine_similarity(v1, v2))
    

Matrix-Vector Multiplication

W = np.array([[0.2, 0.8], [0.6, 0.4]])
x = np.array([1, 2])
output = np.dot(W, x)
print(output)
    

Important Vector Concepts for AI (Intermediate Level)

Unit Vector / Normalization

Used to scale a vector so its length becomes 1. Useful in many AI models to normalize data.

v = np.array([3, 4])
unit_v = v / np.linalg.norm(v)
print("Unit Vector:", unit_v)
   

Angle Between Vectors

Helps measure direction difference between vectors. Closer angle = more similar direction.

def angle_between(v1, v2):
    cos_theta = np.dot(v1, v2) / (np.linalg.norm(v1) * np.linalg.norm(v2))
    angle = np.arccos(cos_theta)
    return np.degrees(angle)

v1 = np.array([1, 0])
v2 = np.array([0, 1])
print("Angle:", angle_between(v1, v2), "degrees")
    

Outer Product

Creates a matrix from two vectors. Used in attention mechanisms in Transformers.

v1 = np.array([1, 2])
v2 = np.array([3, 4])
outer = np.outer(v1, v2)
print("Outer Product:
", outer)
    

Cross Product (3D only)

Used in physics and robotics for direction. Only works with 3D vectors.

v1 = np.array([1, 0, 0])
v2 = np.array([0, 1, 0])
cross = np.cross(v1, v2)
print("Cross Product:", cross)
    

Vector Projection

Used to “shadow” one vector onto another. Great for AI feature extraction.

def projection(a, b):
    return (np.dot(a, b) / np.dot(b, b)) * b

a = np.array([3, 4])
b = np.array([2, 0])
print("Projection of a onto b:", projection(a, b))
   

Book Recommendations

• Math Adventures with Python – Peter Farrell
• Coding Math – Keith Peters
• AI + Machine Learning for Kids – Dale Lane
• Super Simple Math (DK)
• Linear Algebra for Beginners – Richard Bronson