>[!summary] Vector length describes the magnitude of the length from a to b. > Its can be proved through pythagorean theorem. > >Generalized Vector length: >$\begin{array}{c} \text{If $\vec{p} = (x_1, x_2\dots dx_n)$ or $\vec{p} \in R^n$} \\ ||\vec{p}|| = \sqrt{x_1^2 + x_2^2 \dots x_n^2} \end{array}$ >[!info]+ Read Time **⏱ 2 mins** # Proving Vector Length & Angles Using Pythagorean Theorem >[!warning] Assumption When proving vector length we will assume the Pythagorean theorem is true without a formal proof of it. > >We will also assume our plane in $\mathbb{R^2}$ as proof, which is also vaild for $\mathbb{R^n}$ Vector length is the length of the vector as described by its components. This is usfuel if we need to find angles under or above of vector. If we have a vector in $\mathbb{R^2}$ such as $\vec{p}$ who is build for components $\vec{p} = \begin{bmatrix} x_1 \\ x_2 \end{bmatrix}$ then by Pythagorean theorem the length of $\vec{p}$ would have to be $||\vec{p}|| = \sqrt{x_1^2 + x_2^2}$ ![[VL_1.png|500]] ## Generalization Using Pythagorean theorem notice that our length is just the vector in each plane so for a generalization the following must be true: $\begin{array}{c} \text{If $\vec{p} = (x_1, x_2\dots dx_n)$ or $\vec{p} \in R^n$} \\ ||\vec{p}|| = \sqrt{x_1^2 + x_2^2 \dots x_n^2} \end{array}$ --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidan-MathMatter) with a star and help others learn more easily. ---