Encontre
Para os vetores abaixo encontre: a norma, o ângulo, o produto interno e o vetor unitário (versor). Para os vetores em \mathbb{R}^{3} encontre o produto externo.
а) \vec{u}=(2,3,1), \vec{v}=(3,-2,0);
b) \vec{u}=(1,2,-1,3), \vec{v}=(0,0,-1,-2);
c) \vec{u}=(2,0,1), \vec{v}=(2,2,-1);
d) \vec{u}=(0,4,2,3), \vec{v}=(0,-1,2,0);
e) \vec{u}=(2,3,4), \vec{v}=(1,0,3);
f) \vec{u}=(0,-1,2,3), \vec{v}=(1,2,-3,-4);
g) \vec{u}=(1,2,-3), \vec{v}=(0,1,-2);
h) \vec{u}=(4,-2,1,3), \vec{v}=(-1,2,5,-4);
i) \vec{u}=(2,0,-4), \vec{v}=(3,2,1);

Passo 1

O enunciado pede, para cada par de vetores \vec u e \vec v, que calculemos:
  1. A norma de cada vetor, definida por
\|\vec w\| = \sqrt{\sum_i w_i^2}.
  1. O produto interno,
\langle \vec u,\vec v\rangle = \sum_i u_i v_i.
  1. O ângulo \theta entre eles, dado por
\cos\theta = \frac{\langle \vec u,\vec v\rangle}{\|\vec u\|\|\vec v\|}, \quad \theta = \operatorname{arccos}\!\Bigl(\frac{\langle \vec u,\vec v\rangle}{\|\vec u\|\|\vec v\|}\Bigr).
  1. Os versores (vetores unitários) \hat u=\vec u/\|\vec u\| e \hat v=\vec v/\|\vec v\|.
  2. Para vetores em \mathbb R^3, também o produto vetorial (externo) \vec u\times\vec v.

Passo 2

a) \vec u=(2,3,1), \vec v=(3,-2,0) em \mathbb R^3 \|\vec u\|=\sqrt{4+9+1}=\sqrt{14}, \|\vec v\|=\sqrt{9+4+0}=\sqrt{13}. \langle \vec u,\vec v\rangle=2\cdot3+3\cdot(-2)+1\cdot0=0.
Logo \theta=\pi/2. \hat u=(2/\sqrt{14},3/\sqrt{14},1/\sqrt{14}), \hat v=(3/\sqrt{13},-2/\sqrt{13},0). \vec u\times\vec v=(2,3,-13).

Passo 3

b) \vec u=(1,2,-1,3), \vec v=(0,0,-1,-2) em \mathbb R^4 \|\vec u\|=\sqrt{1+4+1+9}=\sqrt{15}, \|\vec v\|=\sqrt{0+0+1+4}=\sqrt{5}. \langle \vec u,\vec v\rangle=1\cdot0+2\cdot0+(-1)(-1)+3(-2)=-5. \cos\theta=-5/(\sqrt{15}\,\sqrt{5})=-1/\sqrt{3}, \theta=\operatorname{arccos}(-1/\sqrt{3}). \hat u=(1/\sqrt{15},2/\sqrt{15},-1/\sqrt{15},3/\sqrt{15}), \hat v=(0,0,-1/\sqrt{5},-2/\sqrt{5}).

Passo 4

c) \vec u=(2,0,1), \vec v=(2,2,-1) em \mathbb R^3 \|\vec u\|=\sqrt{4+0+1}=\sqrt{5}, \|\vec v\|=\sqrt{4+4+1}=3. \langle \vec u,\vec v\rangle=2\cdot2+0\cdot2+1\cdot(-1)=3. \cos\theta=3/(3\sqrt{5})=1/\sqrt{5}, \theta=\operatorname{arccos}(1/\sqrt{5}). \hat u=(2/\sqrt{5},0,1/\sqrt{5}), \hat v=(2/3,2/3,-1/3). \vec u\times\vec v=(-2,4,4).

Passo 5

d) \vec u=(0,4,2,3), \vec v=(0,-1,2,0) em \mathbb R^4 \|\vec u\|=\sqrt{0+16+4+9}=\sqrt{29}, \|\vec v\|=\sqrt{0+1+4+0}=\sqrt{5}. \langle \vec u,\vec v\rangle=4\cdot(-1)+2\cdot2=0. \theta=\pi/2. \hat u=(0,4/\sqrt{29},2/\sqrt{29},3/\sqrt{29}), \hat v=(0,-1/\sqrt{5},2/\sqrt{5},0).

Passo 6

e) \vec u=(2,3,4), \vec v=(1,0,3) em \mathbb R^3 \|\vec u\|=\sqrt{4+9+16}=\sqrt{29}, \|\vec v\|=\sqrt{1+0+9}=\sqrt{10}. \langle \vec u,\vec v\rangle=2\cdot1+3\cdot0+4\cdot3=14. \cos\theta=14/(\sqrt{29}\,\sqrt{10}), \theta=\operatorname{arccos}\bigl(14/\sqrt{290}\bigr). \hat u=(2/\sqrt{29},3/\sqrt{29},4/\sqrt{29}), \hat v=(1/\sqrt{10},0,3/\sqrt{10}). \vec u\times\vec v=(9,-2,-3).

Passo 7

f) \vec u=(0,-1,2,3), \vec v=(1,2,-3,-4) em \mathbb R^4 \|\vec u\|=\sqrt{0+1+4+9}=\sqrt{14}, \|\vec v\|=\sqrt{1+4+9+16}=\sqrt{30}. \langle \vec u,\vec v\rangle=-1\cdot2+2\cdot(-3)+3\cdot(-4)=-20. \cos\theta=-20/(\sqrt{14}\,\sqrt{30})=-10/\sqrt{105}, \theta=\operatorname{arccos}(-10/\sqrt{105}). \hat u=(0,-1/\sqrt{14},2/\sqrt{14},3/\sqrt{14}), \hat v=(1/\sqrt{30},2/\sqrt{30},-3/\sqrt{30},-4/\sqrt{30}).

Passo 8

g) \vec u=(1,2,-3), \vec v=(0,1,-2) em \mathbb R^3 \|\vec u\|=\sqrt{1+4+9}=\sqrt{14}, \|\vec v\|=\sqrt{0+1+4}=\sqrt{5}. \langle \vec u,\vec v\rangle=2+6=8. \cos\theta=8/(\sqrt{14}\,\sqrt{5})=8/\sqrt{70}, \theta=\operatorname{arccos}(8/\sqrt{70}). \hat u=(1/\sqrt{14},2/\sqrt{14},-3/\sqrt{14}), \hat v=(0,1/\sqrt{5},-2/\sqrt{5}). \vec u\times\vec v=(-1,2,1).

Passo 9

h) \vec u=(4,-2,1,3), \vec v=(-1,2,5,-4) em \mathbb R^4 \|\vec u\|=\sqrt{16+4+1+9}=\sqrt{30}, \|\vec v\|=\sqrt{1+4+25+16}=\sqrt{46}. \langle \vec u,\vec v\rangle=4\cdot(-1)+(-2)\cdot2+1\cdot5+3\cdot(-4)=-15. \cos\theta=-15/(\sqrt{30}\,\sqrt{46}), \theta=\operatorname{arccos}\bigl(-15/\sqrt{1380}\bigr). \hat u=(4/\sqrt{30},-2/\sqrt{30},1/\sqrt{30},3/\sqrt{30}), \hat v=(-1/\sqrt{46},2/\sqrt{46},5/\sqrt{46},-4/\sqrt{46}).

Passo 10

i) \vec u=(2,0,-4), \vec v=(3,2,1) em \mathbb R^3 \|\vec u\|=\sqrt{4+0+16}=2\sqrt{5}, \|\vec v\|=\sqrt{9+4+1}=\sqrt{14}. \langle \vec u,\vec v\rangle=6-4=2. \cos\theta=2/(2\sqrt{5}\,\sqrt{14})=1/\sqrt{70}, \theta=\operatorname{arccos}(1/\sqrt{70}). \hat u=(1/\sqrt{5},0,-2/\sqrt{5}), \hat v=(3/\sqrt{14},2/\sqrt{14},1/\sqrt{14}). \vec u\times\vec v=(8,-14,4).

Resposta

a) \|\vec u\|=\sqrt{14}, \|\vec v\|=\sqrt{13}, \langle\vec u,\vec v\rangle=0, \theta=\pi/2, \hat u=(2/\sqrt{14},3/\sqrt{14},1/\sqrt{14}), \hat v=(3/\sqrt{13},-2/\sqrt{13},0), \vec u\times\vec v=(2,3,-13).
b) \|\vec u\|=\sqrt{15}, \|\vec v\|=\sqrt{5}, \langle\vec u,\vec v\rangle=-5, \theta=\operatorname{arccos}(-1/\sqrt{3}), \hat u=(1/\sqrt{15},2/\sqrt{15},-1/\sqrt{15},3/\sqrt{15}), \hat v=(0,0,-1/\sqrt{5},-2/\sqrt{5}).
c) \|\vec u\|=\sqrt{5}, \|\vec v\|=3, \langle\vec u,\vec v\rangle=3, \theta=\operatorname{arccos}(1/\sqrt{5}), \hat u=(2/\sqrt{5},0,1/\sqrt{5}), \hat v=(2/3,2/3,-1/3), \vec u\times\vec v=(-2,4,4).
d) \|\vec u\|=\sqrt{29}, \|\vec v\|=\sqrt{5}, \langle\vec u,\vec v\rangle=0, \theta=\pi/2, \hat u=(0,4/\sqrt{29},2/\sqrt{29},3/\sqrt{29}), \hat v=(0,-1/\sqrt{5},2/\sqrt{5},0).
e) \|\vec u\|=\sqrt{29}, \|\vec v\|=\sqrt{10}, \langle\vec u,\vec v\rangle=14, \theta=\operatorname{arccos}(14/\sqrt{290}), \hat u=(2/\sqrt{29},3/\sqrt{29},4/\sqrt{29}), \hat v=(1/\sqrt{10},0,3/\sqrt{10}), \vec u\times\vec v=(9,-2,-3).
f) \|\vec u\|=\sqrt{14}, \|\vec v\|=\sqrt{30}, \langle\vec u,\vec v\rangle=-20, \theta=\operatorname{arccos}(-10/\sqrt{105}), \hat u=(0,-1/\sqrt{14},2/\sqrt{14},3/\sqrt{14}), \hat v=(1/\sqrt{30},2/\sqrt{30},-3/\sqrt{30},-4/\sqrt{30}).
g) \|\vec u\|=\sqrt{14}, \|\vec v\|=\sqrt{5}, \langle\vec u,\vec v\rangle=8, \theta=\operatorname{arccos}(8/\sqrt{70}), \hat u=(1/\sqrt{14},2/\sqrt{14},-3/\sqrt{14}), \hat v=(0,1/\sqrt{5},-2/\sqrt{5}), \vec u\times\vec v=(-1,2,1).
h) \|\vec u\|=\sqrt{30}, \|\vec v\|=\sqrt{46}, \langle\vec u,\vec v\rangle=-15, \theta=\operatorname{arccos}(-15/\sqrt{1380}), \hat u=(4/\sqrt{30},-2/\sqrt{30},1/\sqrt{30},3/\sqrt{30}), \hat v=(-1/\sqrt{46},2/\sqrt{46},5/\sqrt{46},-4/\sqrt{46}).
i) \|\vec u\|=2\sqrt{5}, \|\vec v\|=\sqrt{14}, \langle\vec u,\vec v\rangle=2, \theta=\operatorname{arccos}(1/\sqrt{70}), \hat u=(1/\sqrt{5},0,-2/\sqrt{5}), \hat v=(3/\sqrt{14},2/\sqrt{14},1/\sqrt{14}), \vec u\times\vec v=(8,-14,4).

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