Vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$
Welcome, students! 👋 In this lesson, you will learn what vectors are in $\mathbb{R}^2$ and $\mathbb{R}^3$, how to describe them, and why they matter in Linear Algebra. By the end, you should be able to explain the key ideas, use vector notation correctly, and connect these vectors to the bigger picture of Euclidean space.
What a vector means in Euclidean space
A vector is a quantity that has both magnitude and direction. In everyday life, a vector can describe a move like “walk $3$ steps east and $2$ steps north” 🚶♂️. In mathematics, we usually represent vectors in coordinate form.
In $\mathbb{R}^2$, a vector has two components:
$$\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}$$
This means the vector moves $x$ units in the horizontal direction and $y$ units in the vertical direction.
In $\mathbb{R}^3$, a vector has three components:
$$\mathbf{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$$
Now the vector includes movement along three perpendicular axes: left-right, forward-back, and up-down.
The notation $\mathbb{R}^2$ means the set of all ordered pairs of real numbers, and $\mathbb{R}^3$ means the set of all ordered triples of real numbers. These are examples of Euclidean spaces, which are the standard spaces used to model geometry in algebra.
Understanding vectors in $\mathbb{R}^2$
A vector in $\mathbb{R}^2$ is written as $\begin{bmatrix} x \\ y \end{bmatrix}$ or sometimes as $(x,y)$. Both forms describe the same object.
For example:
$$\mathbf{u} = \begin{bmatrix} 4 \\ -1 \end{bmatrix}$$
This means move $4$ units to the right and $1$ unit down. Another vector might be
$$\mathbf{w} = \begin{bmatrix} -2 \\ 5 \end{bmatrix}$$
which means move $2$ units to the left and $5$ units up.
You can think of vectors in $\mathbb{R}^2$ as arrows on a grid 📈. The starting point usually does not matter when we talk about the vector itself; what matters is its length and direction. That is why the same vector can be drawn in different places as long as the arrow points the same way and has the same size.
Vector addition in $\mathbb{R}^2$
If
$$\mathbf{u} = \begin{bmatrix} x_1 \\ y_1 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} x_2 \\ y_2 \end{bmatrix},$$
then their sum is
$$\mathbf{u} + \mathbf{v} = \begin{bmatrix} x_1 + x_2 \\ y_1 + y_2 \end{bmatrix}.$$
Example:
$$\begin{bmatrix} 2 \\ 3 \end{bmatrix} + \begin{bmatrix} -1 \\ 4 \end{bmatrix} = \begin{bmatrix} 1 \\ 7 \end{bmatrix}.$$
This is like combining two movements: first go $2$ right and $3$ up, then go $1$ left and $4$ up. The final result is $1$ right and $7$ up.
Scalar multiplication in $\mathbb{R}^2$
A scalar is just a real number. If $c$ is a scalar and
$$\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix},$$
then
$$c\mathbf{v} = \begin{bmatrix} cx \\ cy \end{bmatrix}.$$
Example:
$$3\begin{bmatrix} 2 \\ -1 \end{bmatrix} = \begin{bmatrix} 6 \\ -3 \end{bmatrix}.$$
This triples the length of the vector while keeping the same direction. If the scalar is negative, the vector points in the opposite direction. For instance,
$$-2\begin{bmatrix} 1 \\ 4 \end{bmatrix} = \begin{bmatrix} -2 \\ -8 \end{bmatrix}.$$
Understanding vectors in $\mathbb{R}^3$
Vectors in $\mathbb{R}^3$ work the same way, but now there are three coordinates:
$$\mathbf{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}.$$
The third coordinate gives movement along a third axis, which is important for describing 3D objects like boxes, airplanes in flight, or the location of a drone in the sky 🚁.
Example:
$$\mathbf{a} = \begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}$$
means move $1$ unit in one direction, $2$ in another, and $3$ in the third. Even though we cannot always draw 3D objects perfectly on paper, the algebra works exactly the same.
Addition and scalar multiplication in $\mathbb{R}^3$
If
$$\mathbf{u} = \begin{bmatrix} x_1 \\ y_1 \\ z_1 \end{bmatrix} \quad \text{and} \quad \mathbf{v} = \begin{bmatrix} x_2 \\ y_2 \\ z_2 \end{bmatrix},$$
then
$$\mathbf{u} + \mathbf{v} = \begin{bmatrix} x_1 + x_2 \\ y_1 + y_2 \\ z_1 + z_2 \end{bmatrix}.$$
And for a scalar $c$,
$$c\mathbf{v} = \begin{bmatrix} cx \\ cy \\ cz \end{bmatrix}.$$
Example:
$$\begin{bmatrix} 1 \\ -2 \\ 0 \end{bmatrix} + \begin{bmatrix} 3 \\ 5 \\ -4 \end{bmatrix} = \begin{bmatrix} 4 \\ 3 \\ -4 \end{bmatrix}.$$
This kind of calculation is essential in physics, computer graphics, engineering, and data science 🌟.
Length and direction: the geometric meaning
The length of a vector is also called its magnitude or norm. For a vector
$$\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}$$
its length is
$$\|\mathbf{v}\| = \sqrt{x^2 + y^2}.$$
For a vector in $\mathbb{R}^3$,
$$\mathbf{v} = \begin{bmatrix} x \\ y \\ z \end{bmatrix},$$
its length is
$$\|\mathbf{v}\| = \sqrt{x^2 + y^2 + z^2}.$$
Example in $\mathbb{R}^2$:
$$\left\|\begin{bmatrix} 3 \\ 4 \end{bmatrix}\right\| = \sqrt{3^2 + 4^2} = \sqrt{25} = 5.$$
Example in $\mathbb{R}^3$:
$$\left\|\begin{bmatrix} 1 \\ 2 \\ 2 \end{bmatrix}\right\| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{9} = 3.$$
Two vectors can have the same length but different directions. For instance,
$$\begin{bmatrix} 2 \\ 1 \end{bmatrix} \quad \text{and} \quad \begin{bmatrix} -2 \\ -1 \end{bmatrix}
$$
have the same length but point in opposite directions.
How vectors fit into Linear Algebra
Linear Algebra studies vectors, matrices, and operations that combine them. Vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$ are the starting point for many deeper ideas.
One major idea is linear combinations. A linear combination of vectors $\mathbf{u}$ and $\mathbf{v}$ looks like
$$a\mathbf{u} + b\mathbf{v},$$
where $a$ and $b$ are scalars.
For example, if
$$\mathbf{u} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 0 \\ 1 \end{bmatrix},$$
then any vector in $\mathbb{R}^2$ can be written as
$$x\mathbf{u} + y\mathbf{v} = \begin{bmatrix} x \\ y \end{bmatrix}.$$
This shows that the standard unit vectors form a foundation for the whole plane.
In $\mathbb{R}^3$, the standard unit vectors are
$$\mathbf{e}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}, \quad \mathbf{e}_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}, \quad \mathbf{e}_3 = \begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}.$$
Any vector $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ can be written as
$$x\mathbf{e}_1 + y\mathbf{e}_2 + z\mathbf{e}_3.$$
This idea connects vectors in Euclidean space to the broader topic of span, basis, and coordinate systems.
Common examples and reasoning skills
Here are some useful ways to think about vectors:
- A GPS location can be modeled with coordinates in a plane or in space.
- A wind report might say the wind moves east and north, which is like a vector in $\mathbb{R}^2$.
- A drone’s movement in the air needs three coordinates, so $\mathbb{R}^3$ is more appropriate.
- In computer animation, vectors help represent position, motion, and direction on a screen or in 3D scenes 🎮.
When solving problems, always ask:
- What space is the vector in: $\mathbb{R}^2$ or $\mathbb{R}^3$?
- What are the components?
- Are you adding vectors, scaling them, or finding length?
- Does the result make sense geometrically?
For example, if a hiker walks
$$\begin{bmatrix} 5 \\ 2 \end{bmatrix}$$
one day and
$$\begin{bmatrix} -1 \\ 3 \end{bmatrix}$$
the next, then the total trip is
$$\begin{bmatrix} 5 \\ 2 \end{bmatrix} + \begin{bmatrix} -1 \\ 3 \end{bmatrix} = \begin{bmatrix} 4 \\ 5 \end{bmatrix}.$$
This is a direct example of combining movements as vectors.
Conclusion
Vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$ are basic building blocks of Linear Algebra. They describe position, movement, and direction in the plane and in space. You learned how to write vectors, add them, multiply them by scalars, and find their lengths. You also saw how these ideas connect to linear combinations and the larger structure of Euclidean space. Understanding these foundations will help you with matrices, systems of equations, subspaces, and many future topics in Linear Algebra.
Study Notes
- A vector in $\mathbb{R}^2$ has the form $\begin{bmatrix} x \\ y \end{bmatrix}$, and a vector in $\mathbb{R}^3$ has the form $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$.
- Vectors represent both magnitude and direction.
- Vector addition is done component by component.
- Scalar multiplication multiplies every component by the same real number.
- The length of $\begin{bmatrix} x \\ y \end{bmatrix}$ is $\sqrt{x^2 + y^2}$.
- The length of $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is $\sqrt{x^2 + y^2 + z^2}$.
- Vectors in $\mathbb{R}^2$ and $\mathbb{R}^3$ are part of Euclidean space, the standard setting for geometry in Linear Algebra.
- The standard unit vectors are $\mathbf{e}_1$, $\mathbf{e}_2$, and $\mathbf{e}_3$.
- Any vector in $\mathbb{R}^2$ or $\mathbb{R}^3$ can be expressed using linear combinations of the standard unit vectors.
- These ideas are used in physics, engineering, graphics, navigation, and data science.