Question

calcule os ângulos dos vetores u=(1,3,-1) e v=(0,3,-1)

Answer 1

O ângulo entre dois vetores u e v do Rn é definido por

$\boxed{\boxed{\theta=cos^{-1}\left(\dfrac{\vec{u}\cdot\vec{v}}{||\vec{u}||||\vec{v}||}\right)}}$
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Achando a norma de u:

$||\vec{u}||=\sqrt{1^{2}+3^{2}+(-1)^{2}}\\\\||\vec{u}||=\sqrt{1+9+1}\\\\||\vec{u}||=\sqrt{11}$

Achando a norma de v:

$||\vec{v}||=\sqrt{0^{2}+3^{2}+(-1)^{2}}\\\\||\vec{v}||=\sqrt{0+9+1}\\\\||\vec{v}||=\sqrt{10}$

Achando o produto interno de u e v:

$\vec{u}\cdot\vec{v}=(1,3,-1)\cdot(0,3,-1)\\\\\vec{u}\cdot\vec{v}=1\cdot0+3\cdot3-1\cdot(-1)\\\\\vec{u}\cdot\vec{v}=0+9+1\\\\\vec{u}\cdot\vec{v}=10$

Então, o cosseno ângulo entre u e v é dado por

$cos(\theta)=\dfrac{\vec{u}\cdot\vec{v}}{||\vec{u}||||\vec{v}||}\\\\\\cos(\theta)=\dfrac{10}{\sqrt{11}\cdot\sqrt{10}}\\\\\\cos(\theta)=\dfrac{10}{\sqrt{110}}$

Racionalizando:

$cos(\theta)=\dfrac{10\sqrt{110}}{\sqrt{110}\sqrt{110}}\\\\\\cos(\theta)=\dfrac{10\sqrt{110}}{110}\\\\\\cos(\theta)=\dfrac{\sqrt{110}}{11}$

Então:

$\boxed{\boxed{\theta=cos^{-1}\left(\dfrac{\sqrt{110}}{11}\right)}}$