Question

Calcule o módulo de V=(2,3,1) e seu versor.

Answer 1

V=(2,3,1)

Módulo do vetor V:

$|V|=\sqrt{2^{2}+3^{2}+1^{2}}=\ \sqrt{4+9+1}=\ \sqrt{14}$


Versor do Vetor V:

$\dfrac{V}{|V|}=\dfrac{\left(2,3,1\right)}{\sqrt{14}}=\left(\dfrac{2}{\sqrt{14}},\dfrac{3}{\sqrt{14}},\dfrac{1}{\sqrt{14}}\right)\\ \\ \\ \dfrac{V}{|V|}=\left(\dfrac{2\cdot \sqrt{14}}{\sqrt{14}\cdot \sqrt{14}},\dfrac{3\cdot \sqrt{14}}{\sqrt{14}\cdot \sqrt{14}},\dfrac{1\cdot \sqrt{14}}{\sqrt{14}\cdot \sqrt{14}}\right)\\ \\ \\ \dfrac{V}{|V|}=\left(\dfrac{2 \sqrt{14}}{14},\dfrac{3 \sqrt{14}}{14},\dfrac{\sqrt{14}}{14}\right)$


Lembrando que o módulo do versor é unitário, ou seja, igual a 1. Vamos verificar:


$|\dfrac{V}{|V|}|=\sqrt{\left(\dfrac{2 \sqrt{14}\left}{14}\right)^{2}+\left(\dfrac{3 \sqrt{14}}{14}\right)^{2}+\left(\dfrac{\sqrt{14}}{14}\right)^{2}}}\\ \\ \\ |\dfrac{V}{|V|}|=\sqrt{\left(\dfrac{4 \cdot 14}{196}\right)+\left(\dfrac{9\cdot 14}{196}\right)+\left(\dfrac{14}{196}\right)}\\ \\ \\ |\dfrac{V}{|V|}|=\sqrt{\left(\dfrac{56}{196}\right)+\left(\dfrac{126}{196}\right)+\left(\dfrac{14}{196}\right)}\\ \\ \\ |\dfrac{V}{|V|}|=\sqrt{\dfrac{56+126+14}{196}}=\ \sqrt{\dfrac{196}{196}}=\ \sqrt{1}=1$