Problemas 8: I

(vxV) .
w =

19 i k &R (rxv) 1 24
=0
0 8
=

+ w
+
=
- -

228
a

20 E Ik = +

0 +

3k (2 +

0g 3k) (+
.
-

i +

2 -

k) =

I 1 I
(1)(-1) (0)(2) (3)( 1) &
-

=
-
+ +
-

2 1
-

(v w . . =

19 I k -
Pe +

0y
+

04 12 -

2 = 0
⑧ 20
0 0 2

20 I K =

3 +
4 +

5k (32 +

1y 3k) (2 y k)
-
. -
+
=

2 7 -2 3(1) +

(2)( 1) -
+

(5)(1) =

1 2 -

I 3 -

2 +

5 =
0

(wxv) .

v =

ac
19 I 6 k -
02 +

14 +

0 19 24 .
=

2 o0
a
M
j (2 34) (22 y 24)
20 K
l = +

08 34 +
+

0y +
:
+
-
=

I 1 I
2(1) (0)(1) 3) 2) 1
-

+
+
-
= -

2 I
-

2
Volumen del paralepipedo
(vxw) &r
*
19 U .
=

(vxw) =

20 X
v .
(vxw) =

v =
(vxw) =

13

17 Pa) 1 1 , ,
1) ,
Pa(1 ,
2
,
17 ,
Ps(1 ,
1
,
2) I

P2D =
(1 1) - +

(2 1) 2
-
+

(1 112
- =

1
P2
8

PP ,
=

/(1)1-1)
(2 1)2 + - =
1 -j Pe
. ⑳

Pi
~

x
a =

0 1 0 , ,
E = 0 0 1
, ,

B
Area triangulo a
x I
=

↑8 P (0 , ,
0 ,
01 ,
P2CO ,
1 ,
2) ,
P3(2 ,
2 , 0)

P2D ,
- a =
0 1
, ,
2) P3D ,
-
5 =

2 2 0
, ,

X
X
Area triangulo
ax5
3
G
=
=

·
P3

5 * k 4 6
a 4y 2k SX
= =
-

x
+ =

p
-

Pa
I 8
0 ⑧
,

zh
59a = +

y +
04b =
-

2 +

4y +

04c =

2 +

2y
+

2k

Area paralepipedo =

a .
(b xc)

A
paralepipedo 110 1
1
. =

140
-

11 =

8 +

0 +

0 -

0 0 - +

2
=

10

22222
Problemas 9:

1(1 ,
6, -7) v
=

(3 = , ) ,
2(1 ,
8 ,
-2) T = -

72 ,
-

By
X =

1 -
7x
X =

P +
2 y
=

8 -
8x
(x y , ,
z) =

(1 ,
6 ,
-

7) +

<(3 = , ) ,
z = -

2x

x =

1 +

3x

y =
6 +

IC
z 7
E
=
-
-

9 (1 2 1) ,
,
(3 5 , 2) ,
,
1 2 1
, ,
+

t(3 ,
5 ,
-

2 -
1
,
2
,
1)
1 2 1
, ,
+

t(2 3 , ,
-

3)
1 2, 1 +

2t 3t -3t =

1 +

2t
,
2 +

3t 1 -

3t
, , , ,

6( -

2
,
b ,
(0 4 5) ( 2 , 2, -2)
3) -

, ,
=
-

(x y , ,
z) (0 2 5) t(- 2 2 2)
=

, ,
+

, ,
-

0 9 . ,
5 +
2t , 2t -2t ,

=-2t , 1 +
2t 5-2t
,

z (I ,
-

,
1) ,
(2 ,, -
I - ,
,
1 +

t) -2 ,, -

= ,
-

I ,
1)
I ,
-

I ,
1 +

f -

2 -

I, +

t ,
-

2 -z
, E -

,
1 + -

2
,
3 ,
-E
, -

,
1 + -

2t , 3t ,
Et
I -

2t ,
-

t +

3t
,
1 -

et

& (10 ,
2, -10) +

t 5 -3 5
,
,

10 2 -10 +

t) 5, -3 , 5- 10 2,
,
-10
, .

10
,
2 -10 ,
+

t(3 ,
-

3 ,
-

5)
10 2 -10 , ,
+

(5t ,
-5t , -

5t)
=(10 +

5t
,
2 -

5t , -

10 -

5t)
33L1X =

f +
t
, y
=
5 +
t
,
z =
-

1 + 2t

(2 x = b +
25
, y
=
11 +

45 ,
z =
-
3 +

4 +

b =
6 +

25 -
(t -

25 =

2)( 1) =

6 +

25 =
-

5 +

t =

11 +

15 -

t -

1s =

b
-

1 +

2t =
-

3 +

5
-

2t -

5 = -

25
=

05 =

E =
-

t -

2 2) -
=
2 ((f +
(- 2)) (5 ( 2))
+ -

(-1 2) 2))
+

, ,

t +

1 =
2 =
(2 3 -5) , ,

t =

2 (2(6 +

2( 2)) (11 ( 2)) , (

,
+ -

-
3 +
( -

2))
=(2 3 , ,
-

5)

3P (1 :

x =

1 +
t , y
=
2 t -

,
z
=

3t
(2 x 2 3 1 5 z 65
y
:
= -
+ =

, ,

1 +

t =
2 -

5 (t +

5
=

1)6 3 +

5 =

1
1 +

3 =

2 -

5 3t -

65 =
0 S
=

1 -

3 =
2 -

1 6t +

65 =

b S = -

2
S = -

2 3t -

65 =

3t =

t 3 =

S = -

xx

* I

X
 Problemas 10: * *

1 P (5 =
1 3) r
=

22 -

3y
+

fi *
, , ,

- X
r . =
2(x) +

( -

3)(y) +

1(z) =

ez Az
x
3y y
2x z
2x 3y *
= =
- +
+
= -

2 3

P =
ax +

by +

cz
+

d =
0
2x -

3y +
1z +

d = 0 d =
-

2(5) +

3)1) f(3)
-
=
-

19

P =

3x -

3y
+

&z -

19 =

2 P =
(1 , 2, 5) ,
5 = 1 ,
-

2 ,
0

&
. F =
ex 2y - +
0z =

L =

ex -

2y +
d 0d =

=
-

f(1) 2(2) 0 ex
+
=
= -

2y =
0

3 P =

(6 ,
10 , -

7) ;
=
-

3 , 0, 3

5 P . = ( 5)x
- +

(0)y +

3(2) 0 =
D -

=
3x +

3z +

d =

d =

5(b) -

3) 1) - =

51 P -

=
5x +

3z +

5) =
0

↑ D (0 0 0)
=

, , ,
=

6, -

1 ,
3

[. 6x 3z 0 P 6x 3z d 0
y y
=
= +
- +
+
-
= =

d =

6(0) (0) 3(0) +

-
=

0 D =
6x -

y
+

3z =

0
7 P,
=

(3 ,
5 ,
2) P2 ,
=

(2 ,
3, 1) ,
P3 =) -

1, -

1 , 1)

AB =
-1 -2 ,
,
-

1 BC =

3 ,
-

1 3 ,

Y AB =
x 1 =
ick =

10 , 6 ,
-

2.
-1 -
2 -

1
-

3 -

1 -

Y .
= 0 4 .
AP A =
3, 5 , 2 P =
x,
y ,
z

10 , 6 -2 x 3 5, z 2 0
y
=
- - -
-

,
,

P =
-

10x +

30 +

by -

18 -

2z +

4 =
0
16 0
=
10x +

by -

2z +
=

8 P CO
. ,
1 ,
0) , PeC0 ,
1
,
1) Ps(1 ,
-3 , 1)

EB =

0 0, 1
,
BC =

1,2 ,
-2

* =

AB x Bi =

=
2, 1 ,
0

- F .
=
0 Y = AP A =

0, 1 0
,
P (x, =

y z) ,

Ep =

x,
y
-

1 ,
z
-

2, 10 ·

x,
y , -1 , z =
0

2x 1 +

02 0 2x 1 =

0
y y
- = -
-
+
+ -
275x -1y -9z 8
=

1
x
1y 3z
+
+
=

* =

⑬ +
c(0+ -

Pi) =
P +
c
=
=
0

N1 =
5 ,
-

1 ,
-

9 N2 = 1, 1
,
3

5 =
N , x N = Ik =

22 -

22y +
20k
-L
12
I

- =
1 -1, ,
1

Six =
0 -

4y
-

9z =

8 4y +

3) 2) -
=
1
Py +

3z =
1 & y =
10
12 5/2
y
=

bz
-
=

z = -
2

x +

&(2) +

3) 2) 1 -

x =
& +

6 -

10

P(0 , , -2) x 1 =
=

10 +

1,= -
c
,
-
2 +

c)
x 2 512 x z 2 2
y
-
=
-
+
=
=

, ,

28L1 =

x +

2y
-

z =
2 Ni =
1 ,
2 ,
-

1
(2 3x 2z 1 N2 3, 1 2
y
+ -
- = =

,
=

I ~k 34 7k
6
5y
=

=
-
-

1 2 -

3 -1 2

x =

0 2 y -

z
=

2
y -

z =

2 32 = 4
2z 1
-zy 1z 1 z 4/3
=
=

y
- + + =

y
+
2(43) =

1
y =
8/3 -

1y 5/3
=

P (0 513 , ,
%3) L1 =

10 +

3x ,
%3 -32 ,
415 -

7x)

X 3x 313 5 z 413 7x
y
= -
-
=
=

-
31 2x -

3y +
27 =
-

7 ; x =
1 +

2t , y
=
2 -

t ,
z =
-

3t

( =
(1 +

27 ,
2 -

t ,
-

3t) Ni =
2 ,3 , 2

2(1 2t) +
-

3(2 t) - +

273t) =
-

2 +

It - 6 +

3t -

6t =

-
7

t =

p(1 +

2( 3) 2 ( ,
-

3) ,
-

3( 3)) =
( 5 -

, 5, 9)

32x y
+

ez 12 x 3 2t 1 6t z 2 1kt
y
+

;
+ -
= - = =
=

, ,

(3 2t) -
+

(1 6t) +
+
1(2 -

Yzt) =
12

12 +

2t =
12 2t 0 =

t 0
=

p (3 1 ,
=

,
2)
 X
Problemas 11: -

1x =
2t +

1 , y
=

t2 +
t
X
- -

eI
t -

3 3 X
-

in
-

y x
=

t 1 , zt 1j 1 t 5
y
-
- -

-
-

#123

I
& 5
-

eq

= te 5

-
en I

· y ti : htt I
2
e -
-

s
-

I in I I 'X
2 &
-

j
-

-
I

- -
-

11 11 I 1 1, 1 1

-
-

-
Problemas 12: E
1 x =
3 -

t2 , y
=
t +
5t , t =

1
X
2( 1) 3
Y
y(t) =
-
+

x (t) 3 (1)"-271) 5

2 x 0(t 2t t 1 t 2
y
- +

i
=
= =

y((t) =
b(2)2 - 1
=
23 =
-

23
x (t) -
(2)2
(-f)

3 X +2 1 t1 t 3
y
+ =
=

;
=

y((t) =
1(3)3 =e(33)(2)(2) =

183
x (t) I

2((3) +

1)"

22 t
=
2

15 x +3 t t
2

y
= -

,
t -

=
2 t = -

2
Para (3t-1) FO * t
=

1 A

y(t) =
2t t = -

x (t) 3t -
1 Pl ,) Pl ,)
Recta horizontal t -
=
Vertical t = -
-

horizontal P(0 0)
,
t 0 =

y((t) 2t 0 t 0
=
=
=

Recta vertical

X(t) =
3t2 -

1 =
03t2 =
1 +2 =

!t =

11
11 3
t
=I
=
=

X = t3 -
t x =
0 t(t -1) t =

0 ,
t =
-1 ,
t =
1
2
0
y
t

y
=
=
 z
16 x 1 t 2t
y
-

= + =

- 2

I(*
-

2 = t 13

= 0
X = +

1 =

= 1+ =
-
8t =
-

t t(t 2) + 0 t= 2
y 2t 0 0
- =
=
- =
=

Y(t)
x (t)
=

()
2t -2 =16t
3t
-

2
2
Para todo 3t
t =
0
O
x'
=3
X =

0 t =
-

2(0 8) y
/
Horizontal (8 -1)
,

,
Vertical (1 0)
0 t 2(1 0) (2 0)
,

t 0
y
= =

,
, , ,

Horizontal
y((t) =
2t 22t 2 -
- =
02t 2 t =
=

1p(8 ,
-

1)
Vertical
04 (1 0)
x(t)
zt 0 t = =
= =

t = 3 -
(1 %8 ,
3)

17 X t 1 t3 3t2
y
= - -
=

Ordenada al origen
x =
t -

1 =
0 + =
1(0 2) ,
-

t 3 t2 0 +(t 3) t 0; t 3( 1
0) (2 0)
y
-
- -
=
=

=
=
, , ,

y((t) =
3t -
6t = 3t2 -
6t
x (t] I
-

12t =
3
X

Horizontal
y((t) 3t2 =
-

6t 0 =
t(3t -6) 0 =
t 0 =

+ 2
=

a al
( -
1 ,
0) ,
(1 ,
-

2) Horizont ,
t
Vertical =
0

aft)
5

a
x (t) 1 = -

- IF0 No hay vertical

Problemas 13: X

*
1 r(t) =
(2 cost) +

(2 sent) +
5t , 02tπ

r(t) =-2 sent 5

+2
cost
+

r(t) = ( 2sent)"
-

+ (2cost)2 +

(5)
*

vi(t) =

-(te
1c0st 3 +
+
=

((sent +cost) +5 =

s(t) =

j 3dt = 3t

s(t)
fdt 3i
=

2 vit) =
(6 sen2t) +

(6cos2t) +

(5t) ,
Oct =
π

vi(t) =

(12 cos 2t) -

(12 sen 2t) +

517

v(t) = (12cos 2 ti) +

(-12 sen 2t)2 +

(5K) 2

r(t) =

144c0s"2t +141se 2t +

25
=

164

viCt) =
13
/Bdt=13π
3r(t) =
ti +

(2/3) + 4 , 0 t 18

vi(t) =
e +

4(x)t"y =

e +
t

(i) (t
(t) y) 1 t (1 t)"
=
+
+
= +
=

181 +

tdt =
2(1 +

t)/ = 18 =

)) = 18
3

& r(t) (2 =
+

t)e -
(t +

1) y +

tR ,
0 <t <
3

vi(t) 2 = -

y
+
4

r(t) =
/(1)2
(1)2 +
=

L =

13 3dt =

3 .

3 =

33
11 r(t) =
(Pcost) +
(Psenty +3t4 ,
02t /1
viCt) =
-

Psenti +costy +3
v(t) =
(-bsent) +

(Icosty)2+ (3K)
r(t) =
-(+
16c0st 9 +
=

25 =
5

5(t) =

(5dt =

5t

s() =

12 v(t) =

(Cost+ Esent) +
(sent-tcosty : /2 = t - π

~(t) =

(-sent + Sent + Ecost) +

(cost-cost + Esent) j
viCt) = t cost +

Esent y

~(t) =

(Acoste) + (tsent) = +
tisent =

serty= t

s(t) =J dt
=
↑22t2π sit) =

(Y -
) =

I I = -
Problemas 14:
+
5 Movimiento en la Circunferencia x
y2 =

r(t) =

(sent) +
(Cost)Y :
t -&
=
"
y T2
~(t) =
cost
-senty
~() cos() -sen()
=

= -
y

· *
X

'

x
*
X
X