Gabarito Lista retas

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3.5 Gabarito - Lista 1 Retas
1. x = 6 + 2t, y = 3− 6t, z = 9− t
2. P (3,6,− 4)
3. a.y = 2x− 14, z = 7
4
x− 16; b. y = −x+ 5, z = −1
3
x+ 2
4. x = −3
2
z +
7
2
, y = 2z
5. Paralela ao eixo x : x = 4 + t, y = −5,z = 3 ou y = −5,z = 3. Paralela
ao eixo y : x = 4, y = −5 + t, z = 3 ou x = 4, z = 3. Paralela ao eixo
z : x = 4, y = −5,z = 3 + t ou x = 4, y = −5.
6. a. θ = cos−1(−1
2
); b. θ = cos−1(
√
66
11
).
7. m = −19
5
8. a. I(2,1,3); b. h = 1, t = −1, I(3,8,12)
9. a. y = −2, z = 4; b. x = 3, z = 1; c. x = 4 + t, y = −1 − t, z = 2; d.
x = 2, y = −3 + 2t, z = 4− t.
10. a.
x− 1
1
=
y + 3
4
=
z − 5
−6 , b. x = 1, z = 5, c. x = 1 + t, y = −3− 2t, z = 5.
11. α = −1
2
, t = 2.
12. I(2,− 3,4)
13. I(1,4,− 2)
14.
√
306
3
u.c.
15. 2
√
2u.c.
16. a.
x− 3
1
=
y + 4
−3 =
z − 4
−2 ; b. (5,− 10,0)
17. Sim, sa˜o colineares.
98
18. a. m = 4, b. m = 4
19. x = 2 + 2t, y = −3 + 4t, z = 3 + 8t
20. x = 3− 3
2
t, y = 3− 1
2
t, z = −2 + t
21. a. x = 3− t, y = 2− t, z = 2− 2t; b. x = −1 + 5
2
t, y = −1
2
t, z = 2− t
22. −→v = (6,− 12,− 12); x = 3 + 6t, y = −2− 12t, z = −12t
23.
−→
AB = (3,0,1),
x+ 1
3
= z−3, y = 1; −→AC = (4,−2,−4), x+ 1
4
=
y − 1
−2 =
z − 3
−4 ;
−−→
BC = (1,− 2,− 5), x− 2 = y − 1−2 =
z − 4
−5
24. 7u.c.
25. x = 1 + t, y = −3− 3t, z = 2 + t
26. x = 4 + t, y = −4 + 35
26
t, z = 2− 37
13
t
27.
x− 2
3
=
y + 3
3
=
z − 4
−4
28.
x− 3
3
= y − 3 = z + 2−2
29. a. Pm(0,
1
2
,1), x = 2t, y =
1
2
,z = 1 + 3t; b. Pm(0,0,0), x = t, y = 0, z = 0
99