What Are Vectors? A Fun and Simple Introduction

Introduction to Vectors

Vectors are fundamental entities in linear algebra and play a crucial role in various fields such as physics, computer science, and engineering. They represent quantities that have both magnitude and direction, making them essential for describing physical phenomena like force, velocity, and acceleration.

To put it in very simple terms, imagine you’re going out and you tell your mom, “I’m going out.” While that’s some information, it’s not enough — she won’t know where exactly to find you.

But if you say, “I’m going to Reya’s house, which is 5 blocks down the street,” then she has everything she needs to locate you. She knows she must go 5 blocks (magnitude) in a downward direction (direction). That’s essentially what a vector does — it tells you how far and in which direction to go.

How to Represent a Vector

A simple way to represent a vector in a 2D coordinate system might look like this:

Vectors are usually denoted by a lowercase letter with an arrow on top. In the example above, we have vector-u, which can be visualized as moving 4 units along the x-axis (let’s call that vector-v) and 2 units along the y-axis (let’s call that vector-w).

In 3D space, a vector might be represented by moving 5 units along the x-axis, 6 units along the y-axis, and 3 units along the z-axis.

Another way to represent a vector is in coordinate form, which provides the same information — how far and in what direction to move along each axis:

Similarly, a 3D vector would have three components:

Operations on Vectors

There are two basic types of operations we can perform on vectors:

Addition

Vector addition is straightforward. Given two vectors, adding them means placing the tail of one vector at the tip of the other — without changing its direction or length.

So, addition is essentially moving in the direction and length of vector-v, and then continuing in the direction and length of vector-w.

This concept is even clearer in one dimension (1D), such as along a straight line:

Here, adding vectors vector-v and vector-w is just adding the numbers 3 and 5. So,

For a more mathematical example:

Then,

So, adding vectors means simply adding their corresponding components. As shown above, this result can also be visualized and confirmed graphically, as shown above.

Scaling

Scaling a vector means multiplying it by a number (called a scalar).

Here’s what happens when vector-v is multiplied by 2 (let’s call that 2vector-v).

If vector-v goes from point A to B then 2vector-v goes from point A to C — it’s twice as long, but in the same direction.

Mathematically, if,

then scaling it twice will be,

Again, this can be visualized on a graph, as shown above.

Multiplying a vector by a value greater than 1 stretches it, while multiplying by a value less than 1 compresses (or “squishes”) it.

But what happens when you multiply by a negative scalar — say, -2?

Let’s calculate it first:

Now let’s plot vector-v, 2vector-v and -2vector-v.

You will see that -2vector-v is just like 2vector-v in length, but it points in the opposite direction.

So, multiplying by a negative scalar both scales the vector and flips its direction.

Conclusion

As we’ll explore in upcoming articles, much of linear algebra revolves around these two basic operations — vector addition and scalar multiplication — and how they transform vectors.

For simplicity, we focused on 2D vectors here, but the same concepts apply in higher dimensions.

These basic ideas, when extended and combined, help computers determine how to display graphics on screen, where to place pixels, and how to animate motion.