Calculo 2 De dos variables_9na Edición - Ron Larson

SECCIÓN 11.5 Rectas y planos en el espacio 807
11.5 11.5 Lines Lines and and Planes Planes in in Space Space 807 807
z
z
z
6
D
6
5
D
5
Q = (3, −1, 4) 3
Q = (3,
(3,
−1,
−1,
4)
4)
3
2
−2 2
−2
−2
1
2
3
4
1
2 1
−1
x
3
4
−1 1
−1
x
−2
−2
−2
2
2
3
3
4
4
5
5
y
y
La distancia del punto Q a la recta es
The The distance between the the point point Q
and and the the
6 line is
2.45.
line is 6 2.45. 2.45.
Figura Figure Figure 11.55 11.55
EJEMPLO 7 Hallar la distancia de un punto a una recta
EXAMPLE 7 Finding the Distance Between a Point and a Line
Hallar la distancia del punto Q3, 1, 4 a la recta dada por
Find the the distance between the the point
Q 3, 3, 1, 1, 4
and and the the line line given by by
x 2 3t, y 2t, y z 1 4t.
x 2 3t, 3t, y 2t, 2t, and and z 1 4t. 4t.
Solución Solution Usando Using los the the números direction de dirección numbers 3, 3, 22,
2, y 4, and and se 4, sabe 4, you you que know un vector that that de a direction
dirección
de la vector recta for for es the the line line is is
u u 3, 2, 3, 3, 4. 2, 2, 4 ..
Vector Direction
Direction de dirección vector
vector de for
for la line
line recta.
Para To To determinar find find a point un on on punto the the line, en line, la let let recta, t se 0
and hace and obtain
t 0 y se obtiene
P P 2, 0, 2, 2, 1. 0, 0, 1 ..
Punto Point
Point sobre on
on la the
the recta. line
line
Así, So, So,
PQ \ PQ \
PQ \ 3 3 2, 1 2 ,, 1 0, 40, 0, 4 1 1 5, 1, 5, 5, 3 1, 1, 3
y se and and puede you you formar can can form el producto the the cross vectorial product
4
i i
j j
k k
PQ \ PQ \
PQ \
u u
5 51
1
3 3
2i 2i 2i11j11j 7k 7k 7k2, 11, 2, 2, 11, 7. 11, 7. 7.
3 32
2
4
Por Finally, último, using usando Theorem el teorema 11.14, 11.14, you you se can can encuentra find find the the que distance la distancia to to be be es
D PQ \
PQ \
u u
D
PQ\ u u
174 174 174
29 29 29
6 6 2.45. 2.45.
Ver figura See
See
Figure
Figure 11.55. 11.55.
11.55.
11-5 11.5 Exercises Ejercicios
See
See
www.CalcChat.com
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for
for
worked-out
worked-out
solutions
solutions
to
to
odd-numbered
odd-numbered
exercises.
exercises.
En In In Exercises los Exercises ejercicios 1 and and 1 y 2, 2, the the la figura figure figure muestra shows shows the the la graph graph gráfica of of de a line una line given given recta
dada by by the the por parametric las ecuaciones equations. paramétricas. (a) (a) Draw Draw an an a) arrow arrow Dibujar on on una the the line line flecha to to
sobre indicate indicate la recta its its orientation. para indicar To To print print su dirección. an an enlarged enlarged b) Hallar copy copy of of las the the coordenadas
de dos puntos, P y Q, en la recta. Determinar el vector PQ \ graph, graph,
go go to to the the website website www.mathgraphs.com. (b) (b) Find Find the the coordinates
.
¿Cuál of two es points, la relación and entre on las the componentes line. del vector the vector y los coeficientes
What What is de is the the t en relationship las ecuaciones between between paramétricas? the the components ¿Cuál es of of la the the razón vector vector de
PQ \
of two points, P
and Q, Q, on the line. Determine the vector PQ \ ..
esta and and relación? the the coefficients c) Determinar of of tt
the the las parametric coordenadas equations? de todos Why Why los puntos is is this this
de true? true? intersección (c) (c) Determine con los the the planos coordinates coordenados. of of any any points points Si la of recta of intersection
no corta
awith uno the de the los coordinate planos coordenados, planes. planes. If If the explicar the line line por does does qué. not not intersect intersect a
plane, explain why.
1.
coordinate
x 1 3t
plane, explain why.
2. x 2 3t
1. 1. yx 21 t3t
3t
2. 2. yx 2 3t 3t
zy 2 5t tt
zy 12
t
z 2 5t 5tz
z 1 tt
z
z
z
x
y
x
y
x
y
y
z
z
In En In Exercises Exercises los ejercicios 3 and and 3 4, y 4, 4, determine determinar whether whether si cada each each punto point point yace lies lies sobre on on the the la
line. recta. line.
3. 3. x 2 t, t, y 3t, 3t, z 4 tt
a) a) 0, 0, 6, 6, 6
b) b) 2, 2, 3, 3, 5
4. x 3
y 7
4.
z 2
2 8
a) a) 7, 7, 23, 23, 0
b) b) 1, 1, 1, 1, 3) 3)
In En In Exercises Exercises los ejercicios 5–10, 5–10, 5 a find find 10, hallar sets sets of of conjuntos (a) (a) parametric de a) ecuaciones equations equations paramétricas
(b) symmetric y b) ecuaciones equations equations of of simétricas the the line line through through de la recta the the point por point el parallel parallel punto pa-
to to
and and
(b)
the ralela the given given al vector vector o or or recta line line (if dado (if possible). possible). (si es posible). (For (For each each (Para line, line, cada write write recta, the the
direction escribir direction los numbers numbers números as as de integers.) integers.) dirección como enteros.)
Point Punto Point
5. 5.
0, 0, 0, 0, 0
0
6. 6.
0, 0, 0, 0, 0
0
7. 7.
2, 2, 0, 0, 33
8. 8. 3, 0, 3, 0, 22
9. 9. 1, 0, 1, 0, 11
x
10. 10.
3, 3, 5, 5, 44
Parallel Paralela Parallel to to a
v 3, 3, 1, 1, 5
v 2, 2, 5 2, 2, 11
v 2i 2i 4j 4j 2k 2k
v 6j 6j 3k 3k
x 3t, 2t, t
3 3t, y 5 2t, z 7 7 t
x 1
y 1
2 z 3
3 2