Vectors

Moving a figure by translation means sliding the figure without rotating it. To describe this movement, we use an arrow \(\vec{u}\) called a vector. To translate all points by a vector, we can use:

• the length of the arrow (magnitude) and the direction (orientation and sense).
• the \(x\)-step and \(y\)-step of the vector (how many units to the right/left and up/down).

These two perspectives on a vector give us two definitions.

A vector \(\Vect{u}\) can be defined:

1. Geometrically, as having a direction (including orientation and sense) and a magnitude (length).
2. Algebraically, as an ordered pair of components \(\begin{pmatrix} \textcolor{colordef}{x-\text{step}}\\\textcolor{colorprop}{y-\text{step}}\end{pmatrix}\).

• The \(x\)-step is positive if the movement is to the right and negative if to the left.
• The \(y\)-step is positive if the movement is upwards and negative if downwards.

\(\quad\quad\quad\)

The vector from \(A\) to \(B\) can be written as:

Let \(A(x_A,\,y_A)\) and \(B(x_B,\,y_B)\) be two points. The components of the vector \(\Vect{AB}\) are:$$\Vect{AB} = \begin{pmatrix}x_B - x_A \\ y_B - y_A\end{pmatrix}$$

Two vectors are said to be equal if they represent the same translation.

Two vectors are equal if they have:

• the same direction (parallel lines),
• the same sense (same orientation),
• and the same magnitude (same length).

\(\textcolor{olive}{\Vect{u}} = \textcolor{olive}{\Vect{v}}\)

Two vectors \(\begin{pmatrix}x\\y\end{pmatrix}\) and \(\begin{pmatrix}x'\\y'\end{pmatrix}\) are equal if \(x = x'\) and \(y = y'\).

Two vectors \(\Vect{AB}\) and \(\Vect{CD}\) are equal if and only if \(ABDC\) is a parallelogram (possibly flattened).

The sum of two vectors \(\vec{u}\) and \(\vec{v}\), written \(\vec{u} + \vec{v}\), is the vector corresponding to performing the translation by \(\vec{u}\) followed by the translation by \(\vec{v}\).

$$\begin{pmatrix}x\\ y\end{pmatrix}+\begin{pmatrix}x'\\ y'\end{pmatrix}=\begin{pmatrix}x+x'\\ y +y' \\ \end{pmatrix}$$

For any points \(A\), \(B\), and \(C\),$$\Vect{AB} + \Vect{BC} = \Vect{AC}$$

Having defined vector addition, we can now state that:

The vector associated with the translation that transforms any point into itself is the null vector, denoted \(\Vect{0}\).$$\Vect{v} + \Vect{0} = \Vect{0} + \Vect{v} = \Vect{v}$$

For any point \(A\):$$\Vect{AA} = \Vect{0}$$

The negative vector of \(\vec{u}\) is the vector with the same direction, the same length, but the opposite sense (orientation). It is denoted by \(-\vec{u}\).

$$-\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}-x\\ -y \\ \end{pmatrix}$$

The vector \(\Vect{BA}\) (from \(B\) to \(A\)) is the negative of the vector \(\Vect{AB}\) (from \(A\) to \(B\)):$$\Vect{BA} = -\Vect{AB}$$

To subtract one vector from another, we simply add its negative (opposite).

The subtraction of vectors \(\Vect{u} - \Vect{v}\) is defined as adding the negative of the second vector:$$\Vect{u}-\Vect{v} = \Vect{u} + (-\Vect{v})$$

Calculate \(\Vect{AB}-\Vect{CB}\).

$$\begin{pmatrix}x\\ y\end{pmatrix}-\begin{pmatrix}x'\\ y'\end{pmatrix}=\begin{pmatrix}x-x'\\ y-y' \\ \end{pmatrix}$$

Let \(k\) be a real number. The product \(k\Vect{v}\) is defined by:$$k\begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}kx\\ ky\end{pmatrix}$$

• \(2\Vect{v}\) has the same direction and sense as \(\Vect{v}\), and is twice as long.
• \(-2\Vect{v}\) has the same direction but the opposite sense, and is twice as long as \(\Vect{v}\).
• If \(0 < k < 1\), the vector \(k\Vect{v}\) has the same direction and sense as \(\Vect{v}\), but is shorter.
• \(0\Vect{v} = \Vect{0}\), the null vector.

By the theorem of Pythagoras,

If \(\Vect{v}=\begin{pmatrix}x\\y\end{pmatrix}\), the magnitude or length of \(\Vect{v}\) is $$\|\Vect{v}\|=\sqrt{x^2+y^2}$$

Calculate the length of \(\Vect{v}=\begin{pmatrix}4\\3\\\end{pmatrix}\)

The distance between points \(A(x_A, y_A)\) and \(B(x_B, y_B)\) in the plane is the magnitude of the vector \(\Vect{AB}\):$$AB = \|\Vect{AB}\| = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2}$$

Calculate the distance between the points \(A(1,\ 2)\) and \(B(5,\ 5)\).

Two vectors \(\Vect{u}\) and \(\Vect{v}\) are said to be colinear if there exists a real number \(k\) such that:$$\Vect{u} = k \Vect{v}$$

• The components of two colinear vectors are proportional.
• The zero vector \(\Vect{0}\) is colinear with every vector \(\Vect{u}\), since \(\Vect{0} = 0 \cdot \Vect{u}\).

• Two lines \(\Line{AB}\) and \(\Line{CD}\) are parallel if and only if the vectors \(\Vect{AB}\) and \(\Vect{CD}\) are colinear.
• Three points \(A\), \(B\), and \(C\) are collinear (lie on the same straight line) if and only if the vectors \(\Vect{AB}\) and \(\Vect{AC}\) are colinear.

Let \(\Vect{u} = \begin{pmatrix} x \\ y \end{pmatrix}\) and \(\Vect{v} = \begin{pmatrix} x' \\ y' \end{pmatrix}\) be two vectors in the plane. The determinant of \(\Vect{u}\) and \(\Vect{v}\), denoted \(\det(\Vect{u},\,\Vect{v})\), is defined as:$$\det(\Vect{u},\,\Vect{v}) =\begin{vmatrix}x & x' \\ y & y'\end{vmatrix}= x \cdot y' - y \cdot x'$$

Two vectors \(\Vect{u}\) and \(\Vect{v}\) are colinear if and only if$$\det(\Vect{u},\,\Vect{v}) = 0$$

Let \(A(1,\ 2)\), \(B(5,\ 4)\), \(C(-1,\ -1)\), and \(D(5,\ 2)\).
Are the lines \(\Line{AB}\) and \(\Line{CD}\) parallel?

In this chapter, we have explored vectors as fundamental tools for describing movement and position in the plane, both geometrically and algebraically. Their strength lies in connecting algebraic operations with geometric intuition, enabling us to model and solve a wide variety of problems in mathematics and science.Vectors are not only essential in geometry but also play a key role in the modern world. In artificial intelligence and data science, for instance, almost all information—images, texts, and even relationships—are encoded as vectors. The landmark paper Attention is All You Need (Vaswani et al., 2017) underlines this fact: modern AI systems, such as OpenAI’s ChatGPT or Grok, manipulate vast arrays of vectors to learn patterns and make predictions. As the authors write:"An attention function can be described as mapping a query and a set of key-value pairs to an output, where the query, keys, values, and output are all vectors."Understanding vectors, therefore, not only builds your geometric and algebraic fluency but also connects you to the mathematical foundations of today's digital and AI-driven world.