Gabarito Lista de Exercícios - Estudo da Reta e do Plano

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Geometria Anal´ıtica e A´lgebra Linear - GA21NB
Professor: Geovani Raulino
Gabarito Lista de Exerc´ıcios 03 - A Reta e o Plano
1. Uma equac¸a˜o da reta e´ r: P = A+ t ~AB ou r: (x, y, z) = (2,−3, 4) + t(−1, 2,−2).
O ponto C pertence a reta, pore´m o ponto D na˜o pertence.
2. r :

x = −1 + 2t
y = 2− 3t
z = 3
3. (a) P = (−1, 6,−10). (b) P = (5
2
, 5
2
,−2). (c) P = (−4, 9,−16).
4. (a) r :

x = 1 + t
y = −1 + 2t
z = 2− 2t
(b) r :

x = 3
y = 1− 3t
z = 4− 2t
5. P = (2, 1, 9)
6. r :

x = 2 + t
y = −1− 3
2
t
z = 4 + 2t
7. (a) P = (5,−5, 8)
(b) P = (−9, 2,−20)
(c) P = (4
3
, −19
6
, 2
3
)
(d) P = ( 7
13
, −36
13
, −12
13
)
8. Respectivamente,
ˆ r :

x = 4 + t
y = −5
z = 3
ˆ r :

x = 4
y = −5 + t
z = 3
ˆ r :

x = 4
y = −5
z = 3 + t
9. (a) 120◦ (b) 150◦
10. (a) n = 1 ou n = 7. (b) n = ±√15
11. (a) m = −7
4
. (b) m = 1 ou m = −3
2
.
12. (a) P = (1, 2,−2). (b) P = (3, 8, 12). (c) coincidentes.
13. (a) m = −3 (b) m = 4.
14. (a) (4, 5, 7) e (0,−3,−1) (b) (17
9
, 7
9
, 25
9
)
e (1,−1, 1).
15. r :

x = t
y = 3t
z = 5
16. (a) P = (1, 3, 2);
(b) k = 1
2
;
(c) P = (2,−4, 2);
(d) k = −12.
17. (a) pi : 2x− 3y − z − 13 = 0;
(b) pi : 12x+ 12y + 6z + 9 = 0;
(c) na˜o existe tal plano, pois a reta na˜o e´
ortogonal ao vetor normal do plano pi1;
(d) pi : x+ y + z − 6 = 0.
18. pi:

x = 1 + h
y = 1 + t
z = −5− 2t+ 3h
19. (a) pi : −3x− 6y − 2z + 7 = 0 e pi:

x = 1− 2t
y = 2t+ h
z = 2− 3t− 3h
(b) pi : −4x− 6y + 2z = 0 e pi:

x = t− h
y = t+ h
z = 5t+ h
(c) pi : 18x+ 12y − 36 = 0 e pi:

x = 2− 4t− 2h
y = 6t+ 3h
z = −1 + 4t+ 5h
(d) pi : z − 3 = 0 e pi:

x = 2− 5t+ 2h
y = 1− 2t+ h
z = 3
20. (a) 60◦; (b) 30◦; (c) 26, 57◦
21. m = 1 ou m = 7.
22. m = −12
23. (a) r :

x = 6
7
− t
7
y = 11
7
+ 11t
7
z = t
; (b) r :

x = 2 + t
2
y = 5
2
+ t
4
z = t
; (c) r :

x = −1− t
y = t
z = 1
24. (a) r :

x = 5 + 2t
y = 2 + t
z = 3 + t
;
(b) P ′ = (−3,−2,−1);
(c) 2
√
6
25. (a) pi : x− y − z − 1 = 0;
(b) pi : 10x− 5y + 6z + 30 = 0;
(c) pi : 2x− 11y − 5z + 49 = 0.
26. (a) 3
√
29 (u.m.)2; (b) 6
√
29
5
u.m.; (c) 12 (u.m.)3
27. (a)
√
117
3
u.m.; (b)
√
6
2
u.m.; (c) 0
28. (a) 7
3
u.m.; (b) 2
√
3 u.m.; (c) 6
√
11
11
u.m.
29. 3
√
6
2
u.m.
30. (a) 3
√
5
5
u.m.; (b)
√
2
2
u.m.;