Vector Operations

Welcome back to your study of linear algebra. In our first lesson, we established the core idea that vectors can be viewed in two complementary ways: as geometric arrows and as numerical lists. We also took a first look at how a scalar can change a vector's scale.

Today, we will build directly on that foundation by mastering the two fundamental operations of vector arithmetic: addition and scalar multiplication. You will learn to perform these operations both graphically, by manipulating arrows, and mathematically, by working with their components. These two operations are the foundation upon which much of linear algebra is built, so a solid understanding here will set you up for success in all the topics to come.

Vector Addition

Let's begin with addition. If vectors represent movements, how would you combine two movements? For instance, if you walk 1 block east and 2 blocks north ( $v_{1}$ ), and then you walk 3 blocks east and 1 block south ( $v_{2}$ ), where do you end up? Vector addition gives us the answer.

The 3Blue1Brown "Essence of linear algebra" series provides an excellent visual explanation of how this works.

Vectors | Chapter 1, Essence of linear algebra

This segment from "Vectors | Chapter 1" demonstrates how to think about vector addition both geometrically and numerically.

Please watch the section explaining vector addition, from the start of the topic. Pay close attention to the tip-to-tail method for adding vectors graphically and how it translates into adding the numbers in the vectors.

As the video so clearly illustrates, there are two ways to think about this operation, and they perfectly correspond with each other.

1. The Graphical Method: Tip-to-Tail

Geometrically, adding two vectors $u$ and $v$ involves a simple procedure:

Start with the first vector, $u$ , rooted at the origin.
Take the second vector, $v$ , and move it so that its tail is at the tip of $u$ .
The sum, $u + v$ , is the new vector that starts at the origin and ends at the tip of the repositioned $v$ .

This is often called the tip-to-tail (or head-to-tail) rule. As the video notes, this is the one time in linear algebra when we allow a vector to move away from the origin, purely for the purpose of visualizing the addition.

An alternative way to visualize the sum, which keeps both vectors at the origin, is the Parallelogram Law. If you draw $u$ and $v$ both starting from the origin, they form two adjacent sides of a parallelogram. The sum $u + v$ is the vector that corresponds to the diagonal of that parallelogram that also starts at the origin.

Parallelogram Law of Vector Addition
The sum of vectors $u$ and $v$ can be found using the Parallelogram Law. The resultant vector $u + v$ is the diagonal of the parallelogram formed by $u$ and $v$ .

Notice that $u + v = v + u$ . This property, known as commutativity, is clear from the parallelogram diagram: it doesn't matter which vector you consider first.

2. The Mathematical Method: Component-wise Addition

Numerically, the process is even simpler. To add two vectors, you add their corresponding components. If $u$ and $v$ are two vectors in $R^{n}$ , their sum is calculated as follows:

u + v = u_{1} u_{2} ⋮ u_{n} + v_{1} v_{2} ⋮ v_{n} = u_{1} + v_{1} u_{2} + v_{2} ⋮ u_{n} + v_{n}

Let's use the example from the video. If $v_{1} = [12]$ and $v_{2} = [3 - 1]$ , then:

v_{1} + v_{2} = [1 + 3 2 + (- 1)] = [41]

This makes perfect sense: the total horizontal movement is $1 + 3 = 4$ , and the total vertical movement is $2 - 1 = 1$ . The math perfectly describes the geometry.

A crucial rule to remember is that you can only add vectors that have the same number of components—that is, they must belong to the same space $R^{n}$ . You cannot add a vector in $R^{2}$ to one in $R^{3}$ .

Scalar Multiplication: A Review

In our last lesson, you were introduced to scalar multiplication. Let's quickly review this second fundamental operation, again with the help of 3Blue1Brown.

Vectors | Chapter 1, Essence of linear algebra

This short clip recaps the concept of scalar multiplication, which we touched on in the previous lesson.

Watch the segment from the start of the scalar multiplication section. This will reinforce the geometric idea of "scaling" and the corresponding numerical operation.

As a reminder:

Geometrically, multiplying a vector $v$ by a scalar $c$ means scaling its length by a factor of $∣ c ∣$ . If $c$ is positive, the direction stays the same. If $c$ is negative, the vector's direction is reversed.
Mathematically, you multiply every component of the vector by the scalar: $c v = c v_{1} v_{2} ⋮ v_{n} = c v_{1} c v_{2} ⋮ c v_{n}$

For example, if $v = [2 - 3]$ and $c = - 2$ , then:

- 2 v = - 2 [2 - 3] = [- 4 6]

Geometrically, the new vector points in the opposite direction of the original and is twice as long.

Combining the Operations: Vector Subtraction

Now we can combine these two operations to define vector subtraction. Subtraction is simply addition of a negative vector.

u - v = u + (- 1) v

Let's break down what this means:

Scale: First, you take the vector $v$ and multiply it by the scalar $- 1$ . This gives you a new vector, $- v$ , which has the same length as $v$ but points in the exact opposite direction.
Add: Then, you add $u$ and the new vector $- v$ using the tip-to-tail method.

Numerically, this translates to component-wise subtraction:

u - v = [u_{1} u_{2}] - [v_{1} v_{2}] = [u_{1} - v_{1} u_{2} - v_{2}]

For example, using our vectors $v_{1} = [12]$ and $v_{2} = [3 - 1]$ :

v_{1} - v_{2} = [1 - 3 2 - (- 1)] = [- 2 3]

Expressions like $c u + d v$ are known as linear combinations. The ability to scale vectors and add them together is arguably the most important idea in this field. Almost everything you do in linear algebra can be seen as a variation on this theme.

Conclusion

In this lesson, we have thoroughly explored the two fundamental operations that form the bedrock of linear algebra.

Here are the key takeaways:

Vector Addition: This can be performed graphically using the tip-to-tail rule or the Parallelogram Law. Mathematically, it is done by adding the corresponding components of the vectors.
Scalar Multiplication: This scales a vector. Geometrically, it stretches, shrinks, and/or reverses the vector. Mathematically, it is done by multiplying every component by the scalar.
Vector Subtraction is not a separate operation, but a combination of scalar multiplication (by -1) and vector addition.

You are now equipped with the essential tools for manipulating vectors. So far, our operations have taken vectors and produced new vectors. In our next lesson, we will introduce the dot product, a fascinating way of "multiplying" two vectors that results not in another vector, but in a single number—a scalar. This will unlock our ability to talk about geometric concepts like length and angle in any dimension.