Vector Addition (The Visual Guide for Developers)

Vector Addition (Visually Explained)

We've covered what a vector is and how to find its length. The next logical step is to figure out what happens when we add two of them together. Luckily, it's simpler than we might think.

This post breaks down the intuition behind vector addition. Watch the video for a quick visual walkthrough, then read on for the details.

The "Head-to-Tail" Rule

The easiest way to understand vector addition is to think of it as taking a journey. Let's say we have two vectors: a = [4, 1] and b = [1, 2].

Adding them, a + b, means we first follow the path of vector a. Then, starting from where a ended (its "head"), we follow the path of vector b (For the below example where b = [1, 2] we follow one unit towards right and 2 units up). The result is a new vector, c, that points from the origin directly to your final destination.

The resulting vector c = [5, 3] is the sum of vectors a and b.

The Math

The math behind the "head-to-tail" rule is incredibly simple. You just add the corresponding elements from each vector.

[4, 1] + [1, 2] = [4+1, 1+2] = [5, 3]

It’s the same element-wise addition you'd expect. It really is that straightforward.

Why We Use NumPy in Python

For vector operations in Python, we almost always use a library called NumPy. This is because standard Python lists don't behave the way we need them to for math.

If you try to add two lists, Python just concatenates them:

# This is NOT vector addition!
a = [4, 1]
b = [1, 2]
c = a + b 

print(c)
# Output: [4, 1, 1, 2]

NumPy, on the other hand, is built for this kind of math and performs the correct element-wise addition automatically:

import numpy as np

a = np.array([4, 1])
b = np.array([1, 2])

# This IS vector addition
c = a + b

print(c)
# Output: [5 3]

Conclusion

Vector addition is a fundamental concept that combines the "journeys" of two or more vectors to find a final result. While the visual "head-to-tail" rule is great for intuition, the math is just simple, element-wise addition—a core operation you'll see everywhere in machine learning.

Your Turn...

What about vector subtraction? How do you think that would work visually on a graph? Share your ideas in the comments!

This post is part of the "Decoding the Math of ML" series. For the previous part, check out: How to Calculate a Vector's Magnitude.

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