>[!summary] Assuming vectors are some form ${\begin{bmatrix} x_1 \\ x_2\end{bmatrix}}$ in a $\mathbb{R^2}$ plane create a point. If we multiply this vector by some scalar, you get a line. We can move this vector in any way to create a new parallel line. > Directed line segments are a way of defining vectors in any reference (changing our reference away from the origin), and are used to define and determine others vectors are equal. > **Key equations:** > Vector equations for parallel lines: $x = \vec{p} + t\vec{d}$ >[!info]+ Read Time **⏱ 3 mins** # Defining Vector Equations for Lines in 2D >[!warning] Assumptions For making lines in $\mathbb{R^2}$ assume [[Vectors Addition & Multiplication|vector addition and multiplication]] are valid. If we have a vector $\vec{d}$ in $\mathbb{R^2}$ this is some vector. If this vector is multiplied by a scalar multiple we can make a line with some direction, which is seen from the figure below. If we have another vector $\vec{p} \in \mathbb{R^2}$ we can make another vector by adding vector $\vec{d}$ and $\vec{p}$. This gives us $\vec{p} + \vec{d}$. If we do the same as before by multiplying $\vec{d}$ by a scalar, we get a new parallel line, seen below, so the key equation here is $ x = \vec{p} + t\vec{d} \quad | t\in \mathbb{R} $ ![[vq_1.png]] >[!note] Explanation If we have some vectors in $\mathbb{R^2}$ and some vector $\vec{d}$ we can do some things to manipulate this vector. > Multiplying this vector creates a line, and we can move the vector and do the same to create a parallel line to the original vector $\vec{d}$ ## Directed Line Segments Sometimes is important to denote a line segment with a direction. This is useful for making geometry statements in reference to some point such as O like in the example below. >[!warning] Assumption In this definition directed line segments for $\mathbb{R^2}$ we assume it in reference with point O on the graph below. But a reference point can be anywhere in a space. We will denote the directed line segments from P to Q by $ \text{Line from P to Q =}\vec{PQ} $ As well our vector $\vec{OP}$ as $\vec{p}$ and $\vec{OQ}$ as $\vec{q}$. ![[LR2_2.png|500]] As well vectors can be equal if they give the same result. Like in the example below, where $\begin{array}{c} \vec{TR}=\vec{QO} \\ \vec{R}-\vec{T}=\vec{Q}-\vec{ O} \\ \\ \end{array} $ ![[VEQ_1.png|500]] --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. ---