Summary  
This chapter introduces the vector data type and demonstrates how to represent 2D vectors and perform operations on them—addition, subtraction, scalar multiplication, magnitude calculation, and the dot product.

General domain of usage  
Computer graphics

**A vector** is a mathematical object that represents both direction and magnitude in space. In data science, vectors are used to describe data points, features, and model parameters such as weights.

Definition

## What Is a Vector?

A vector is an ordered pair of numbers with both **magnitude** and **direction**.

$$
\vec{v} = (x,y)
$$

Vectors are often drawn as arrows from the origin to a point in space. Two vectors are considered equal if they have the same direction and length, even if they start at different locations.



## The Zero Vector

The zero vector has no length and no direction. It is written as:

$$
\vec{0} = (0, 0)
$$


## Vector Addition and Subtraction

### Addition
To add two vectors, add their corresponding components:

$$
\vec{a} + \vec{b} = (a_1 + b_1, \; a_2 + b_2)
$$

You can visualize this with:
- **Head-to-tail method**: move the tail of one vector to the head of the other;
- **Parallelogram method**: both vectors start from the same point and form a parallelogram.



### Subtraction
To subtract one vector from another:

$$
\vec{a} - \vec{b} = (a_1 - b_1, \; a_2 - b_2)
$$

This gives a new vector pointing from the head of the second to the head of the first.

## Scalar Multiplication

Multiplying a vector by a number (a scalar) stretches or flips the vector:

$$
k \cdot \vec{a} = (k \cdot a_1, \; k \cdot a_2)
$$

- If $$k > 1$$, the vector is stretched in the same direction;  
- If $$0 < k < 1$$, the vector is shrunk;
- If $$k < 0$$, it flips direction;
- If $$k = 0$$, it becomes the zero vector.

## Vector Magnitude (Length)

The magnitude or length of a vector is calculated with the **Pythagorean theorem**:

$$
|\vec{a}| = \sqrt{a_1^2 + a_2^2}
$$

This gives the straight-line distance from the origin to the tip of the vector.

## The Dot Product

The **dot product** combines two vectors into a single number that reflects how aligned they are:

$$
\vec{a} \cdot \vec{b} = a_1 b_1 + a_2 b_2
$$

- If the result is **positive**: the vectors point in a similar direction;
- If the result is **zero**: the vectors are perpendicular;
- If the result is **negative**: they point in opposite directions.  

### Example
If $$ \vec{a} = (1, 2)\ \ \text{and}\ \ \vec{b} = (3, 4)$$, then:

$$
\vec{a} \cdot \vec{b} = 1 \cdot 3 + 2 \cdot 4 = 11
$$

If $$\vec{a} = (1, 0),\ \vec{b} = (0, 1)$$. Then their dot product is:

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

Introductions to Vectors

What Is a Vector?

The Zero Vector

Vector Addition and Subtraction

Addition

Subtraction

Scalar Multiplication

Vector Magnitude (Length)

The Dot Product

Example