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Lesson 1: What is a Vector?

1. Introduction to Vectors

A vector is a mathematical object that has both magnitude (size or length) and direction.

Examples from real life:

In mathematics, we usually represent vectors as arrows or lists of numbers.

2. How to Write a Vector

In 2D (flat plane), a vector is written as:

\[ \vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix} \quad \text{or} \quad \vec{v} = (3, 4) \]

The first number is the x-component (horizontal), the second is the y-component (vertical).

In 3D (space), it has three components:

\[ \vec{v} = \begin{pmatrix} 1 \\ -2 \\ 5 \end{pmatrix} \]

Exercise 1

Write the components of the vector \[ \vec{a} = \begin{pmatrix} x \\ y \end{pmatrix} \] that points 5 right and 12 up:
x = ,
y =

Exercise 2

Which vector points 2 units left and 3 units down?
\( \vec{b}_{x} = \) , \( \vec{b}_{y} = \)

3. Summary – Lesson 1

Key ideas:

Next lesson: How to add, subtract, and scale vectors.

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