>[!summary] >Vectors are in $\mathbb{R^3}$ when the set of vectors are in the form $\begin{bmatrix} x_1 \\ x_2 \\ x_3\end{bmatrix} | x_1, x_2, x_3 \in \mathbb{R}$ Finding parallel lines in a 3-dimensions to a point exist as a plane. > **Key equations:** > Equations to describe parallel lines in 3 dimensions to a point P. >$\begin{array}{c} \vec{x} = \vec{p} + t\vec{d} + s\vec{v} | s,t \in \mathbb{R} \end{array}$ >[!info]+ Read Time **⏱ 2 mins** # Vectors in 3D Vectors in $\mathbb{R^3}$ are any set of vectors in the form $\begin{bmatrix} x_1 \\ x_2 \\ x_3\end{bmatrix} | x_1, x_2, x_3 \in \mathbb{R}$. A good way of thinking about vectors in $\mathbb{R^3}$ are that they have 3 perpendicular axis, where any point has a vector in all three axes. If vectors follow these requirements they follow the same conditions and rules from [[Vectors, Vectors Addiction & Vector Multiplication]] and [[Vectors Equations of a Line in 2D]]. # Vector Planes in 3D >[!warning] Assumptions For this description assume the following graph depicts vectors in $\mathbb{R^3}$ in the x,y,z plane, but this is valid for any depiction of $\mathbb{R^3}$ ![[R3_1.png|400]] >[!note] Explanation A plane with a point P in $\mathbb{R^3}$ Using the graph above if we wanted to find all possible points to find parallel lines with point P like we did in [[Vectors Equations of a Line in 2D]]. **Notice how there are two ways a line could be parallel with point P.** Our new definitions needs to take into account this possibility. >[!warning] Derivations Assumptions If we are looking for parallel lines with point P. This can happen with lines coming from the x and y-axis, but not z axis since these lines would be perpendicular. > So we will assume the following: $t, s$ are scalar multiples We will let all the possible parallel lines in the “x” and “y” be denoted by the following: $\begin{array}{c} \vec{x} =t\vec{d} + s\vec{v} | s,t \in \mathbb{R} \end{array}$ Which would describes the plane at origin and like how we did in [[Vectors Equations of a Line in 2D]] to describes the plane parallel to point P we do the following: $\begin{array}{c} \vec{x} = \vec{p} + t\vec{d} + s\vec{v} | s,t \in \mathbb{R} \end{array}$ --- > 📚 Like this note? [Star the GitHub repo](https://github.com/rajeevphysics/Obsidan-MathMatter) to support the project and help others discover it! ---