28 Feb

## 2D Vector Problems Solved using Calcuhubs Calculator App

A 2D Vector is a vector geometry in 2-dimensions and can be calculated by taking the square root of the sum of each components in space.

$\dpi{100}&space;\LARGE&space;\cos\theta=\frac{\left(\overrightarrow{a\&space;\&space;}.\&space;\&space;\overrightarrow{b}\right)}{\left|\overrightarrow{a}\right|\&space;\&space;\left|\overrightarrow{b}\right|}$

## Formula of Angle Between 2D Vectors

$\dpi{100}&space;\LARGE&space;\theta=\cos^{-1}\left[\frac{\left(x_a\times&space;x_b\right)+\left(y_a\times&space;y_b\right)}{\left(\sqrt{\left(\left(x_a\right)^2+\left(x_b\right)^2\right)}\right)\times\sqrt{\left(\left(y_a\right)^2+\left(y_b\right)^2\right)}}\right]$

## Problem 1. Angle Between Two Vectors

Solve the angle between these two vectors  $\dpi{100}&space;\LARGE&space;a\left(12,\&space;7\right),\&space;b\left(5,\&space;6\right)$.

### Solution

$\dpi{100}&space;\LARGE&space;\left(\overrightarrow{a\&space;\&space;}.\&space;\&space;\overrightarrow{b}\right)$

$\dpi{100}&space;\LARGE&space;a.b=\sum_{i=1}^na_ib_i=a_1b_1+a_2b_2+...+a_nb_n$

$\dpi{100}&space;\large&space;a.b=\sum_{i=1}^na_ib_i=a_i=12,\&space;b_i=5,a_j=7,\&space;b_j=6$

$\dpi{100}&space;\LARGE&space;(12\times5)+(7\times6)$

$\dpi{100}&space;\LARGE&space;(60)+(42)=102$

$\dpi{100}&space;\large&space;\dpi{100}&space;\LARGE&space;\cos\theta=\frac{102}{\sqrt{\left(12^2+7^2\right)\times\left(5^2+6^2\right)}}$

$\dpi{100}&space;\large&space;\dpi{100}&space;\LARGE&space;\cos\theta=\frac{102}{\sqrt{\left(144+49\right)\times\left(25+36\right)}}$

$\dpi{100}&space;\large&space;\dpi{100}&space;\LARGE&space;\cos\theta=\frac{102}{\sqrt{\left(193\right)\times\left(61\right)}}$

$\dpi{100}&space;\large&space;\dpi{100}&space;\LARGE&space;\cos\theta=\frac{102}{\sqrt{\left(11773\right)}}$

$\dpi{100}&space;\LARGE&space;\theta=\cos^{-1}\left(\frac{102}{108.5034}\right)$

#### Representing vector result in degree

$\dpi{100}&space;\LARGE&space;\theta=\cos^{-1}\left(0.94\right)$

$\dpi{100}&space;\huge&space;\theta=19.938^{\degree}$

#### Representing vector result in radians, first convert the degree to radian as shown below

$\dpi{100}&space;\LARGE&space;1^{\degree}\times\frac{\pi}{180}=1Rad$

#### Since you have 19.938 degree, then in radians will be

$\dpi{100}&space;\LARGE&space;19.938^{\degree}\times\frac{\pi}{180}$

$\dpi{100}&space;\LARGE&space;19.938^{\degree}\times\frac{3.142}{180}$

$\dpi{100}&space;\LARGE&space;19.938^{\degree}\times0.0174555556$

$\dpi{100}&space;\huge&space;=&space;0.3480$ Radians