EXERCICIOS CVGA - 217 a 226

Prévia do material em texto

EXERCICIOS – CÁLCULO VETORIAL E GEOMETRIA ANALITICA 
 
 
217. A= ( 1 , 1 ,-2 ) – Perpendicular ao plano 
 B= ( 1 , 2 , 3 ) 
 C= ( 4 , -1 , 5) 
 D= ( 2 , -1, 0 ) 
 
( C – D ) = ( 2 ,-1 , 0 ) – ( 4,-1 , 5 ) = ( -2 , 0 , -5) 
( C – B ) = ( 4 ,-1 ,5 ) – ( 1 ,2 , 3 ) = ( 3, -3 , 2) 
 
I J K I J 
3 -3 2 3 -3 
-2 0 -5 -2 0 
 
= 15 i - 4j + 0k + 15j + 0i – 6k = (15 i , 11j , -6k ) 
 
R: ( 1 ,1 , -2 ) + t ( 15, 11 ,-6) 
218. 
 : x + by -2z – 6 = 0 
r: x = a + t 
 y = 2 – t 
 z = -1 + 2t 
 
Wα = ( 1 , b , - 2 ) 
 
r: ( a , 2 , -1 ) + t ( 1 ,-1 , -2 ) 
 
u ∙ Wα = ( 1 , - 1 , 2 ) ∙ ( 1 , b , -2 ) = 0 
1-1b - 4 = 0 
 
b = -3 
 
 : x + by -2z – 6 = 0 
a - 3 . 2 - 2.( -1) – 6 =0 
a – 6+ 2 – 6 =0 
a = 10 
219. 2X – Y + 3Z +3 = 0 ( -1 ) 
 3X – Y + 2Z -1 = 0 
 
 
-2X + Y - 3Z – 3 = 0 
 3X – Y + 2Z - 1 = 0 
 
1X – 1Z – 4 = 0 
X = Z + 4 
 
Substituir o x na equação 
 
- 2 .( Z+ 4 ) + Y – 3Z – 3 = 0 
-2Z – 8 + Y – 3Z – 3= 0 
-5Z – 11 +Y = 0 
Y = 5Z + 11 
 
Paramétricas 
 
 X: 4 +t 
 Y: 11 +5t 
 Z: t 
 
 
R: ( 4 , 11 , 0 ) + t ( 1 , 5 , 1 ) 
220. 
P = ( 1 , - 3 , 5 ) 
 
Produto Vetorial 
 
 
I J K I J 
3 -4 1 3 -4 
4 -7 3 4 -7 
 
- 12 i + 4j – 21k +7i – 9j + 16k = ( -5i , -5j, -5k ) 
 
R: ( 1 ,-3 , 5 ) + t ( -5, -5 , -5 ) 
 
221. 
r: x = ( 1 , -1 , 0 ) + t ( 1 , 3 , 1) 
α: 4x – 3y – z + 2 = 0 
 
Equação vetorial do plano 
 
( x , y , z ) = ( 1 , -1 , 0 ) + t1 ( 1 , 3 ,1 ) + t2 ( 4 ,-3 , -1 ) 
 
 
( x – 1) ( y +2 ) z ( x- 1) ( y +2 ) 
1 3 1 1 3 
4 -3 -1 4 -3 
 
- 3 ( x - 1 ) + 4 ( y + 1 ) -3z + ( y +1 ) + 3 ( x- 1 ) -12z 
4y + 4 – 3z + y + 1 -12z 
5y + 5 - 15z = 0 ( divide por 5) 
 
R: y + 5 -3z = 0 
 
222. 
 
A = ( 2 , - 4 , 1) + t ( 3 , 1 , -1 ) 
X = ( 2 , -5 , 3 ) + t ( 1 , 3 , 1) 
 
3x + y - z – 1 = 0 
 
 
( x – 2) ( y +4 ) ( z - 1 ) ( x- -2) ( y +4 ) 
3 1 -1 3 1 
1 3 1 1 3 
 
1 ( x – 2) - 1 ( y +4 ) + 9 ( z -1) - 3 ( y + 4 ) + 3 ( x-2 ) - 1 ( z-1 ) 
x- 2- y – 4 + 9z -9 – 3y -12 + 3x – 6 – z +1 
4x – 4y + 8z -32 = 0 ( divide por 4 ) 
 
R: x – y + 2z – 8 = 0 
223. 
P = ( 2 , -1 , 1 ) * Planos paralelos tem a mesma direção normal 
α: x – 2y + 4z + 1 = 0 
 
Equação geral 
 
 1x – 2y + 4z + d = 0 
1.2 – 2.-1 + 4.1 + d = 0 
2 + 2 + 4 + d = 0 
d = - 8 
 
Escolher pontos que zerem o d da equação: 
Ponto A = ( 0 , 0 , 2) -> ( P – A ) = ( 2 , - 1 ,1 ) – ( 0 ,0 ,2 ) = ( 2 ,- 1, - 1 ) 
Ponto B = ( 0 ,- 4 , 0 ) -> ( B – A) = ( 0 ,- 4 , 0 ) – ( 0 , 0 ,2 ) = ( 0, -4 , -2 ) vetor não // 
 
 
( x, y , z ) = ( 2 ,- 1, 1 ) + t1 ( 2 ,- 1 , - 1 ) + t2 ( 0 ,-4 , -2 ) 
 
Paramétricas 
 
 
R: 
 X : 2 + 2t1 
 Y : - 1- 1t1 – 4t2 
 Z: 1 -1t1 – 2t2 
 
224. 
 
A = ( -3 , 1 , 2 ) 
 
X= ( 1 , 1 ,1 ) + t ( 4 , -1 , 5 ) 
 
Equação geral do plano 
 
4x – 1y + 5z + d = 0 
4.-3 -1.1 + 5.2 + d = 0 
-12-1 +10 + d = 0 
-13 +10 + d = 0 
 d= 3 
 
R: 4x – 1 y + 5z + 3 = 0 
 
 
 
 
 
 
 
 
 
225. 
Ponto A = ( 1 , 2 , 3 ) 
Plano ( 3 ,- 1 , 1 ) ( x, y, z ) 
 
 R: X – 1 /3 = Y- 2/1 = Z -3 
 
 
 
226. 
A = ( 3 ,-1 , 4 ) 
 
X + 2Y + Z+ 8 = 0 
2X + Y + Z – 4 = 0 
 
( x – 2) ( y +4 ) ( z - 1 ) ( x- -2) ( y +4 ) 
3 1 -1 3 1 
1 3 1 1 3 
2 i + 2j + 1k – 1j – 1i – 4k 
( 1i , 1j, - 3k ) 
 
 
R: ( 3,-1,4 ) + t ( 1 , 1 ,-3 )