Pract. Dirigida #5-6

7/24/2019 Pract. Dirigida #5-6

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PRCTICA DIRIGIDA N 5 6

Demostrar las siguientes proposiciones

1. u x v es ortogonal a u y v

Solucin:

(u x v). u = 0

| i j ku1 u2 u3v1

v2

v3

| . u = [(u2v3 u3v2)! (u3v1 u1v3)! (u1v2 u2v1)".(u1!u2! u3)

= u1u2v3 u1u3v2# u2u3v1 u1u2v3# u1u3v2

u2u3v1 = 0

(u x v). v = 0

| i j ku1 u2 u3v1

v2

v3

| . v = [(u2v3 u3v2)! (u3v1 u1v3)! (u1v2 u2v1)".(v1!v2! v3)

= u2v1v3 u3v1v2# u3v1v2 u1v2v3# u1v2v3u2v1v3= 0

2. $ x (u # v) = $ x u # $ x v

% &

(%) =$ x (u # v) =

|

i j k

w1

w2

w3

u1+v1 u1+v2 u3+v3

|= ($2u3# $2v3 $3u1 $3v2)! ($3u1# $3v1 $1u3 $1v3)!

($1u1# $1v2 $2u1 $2v1)


2/9

(&) = $ x u # $ x v = |i j k

w1

w2

w3

u1

u2

u3| # |

i j k

w1

w2

w3

v1

v2

v3|

= [($2u3 $3u2)! ($3u1 $1u3)! ($1u2 $2u1)" # [($2v3 $3v2)!

($3v1 $1v3)!($1v2 $2v1)"= ($2u3# $2v3 $3u1 $3v2)! ($3u1# $3v1 $1u3 $1v3)!

($1u1# $1v2 $2u1 $2v1)

(%) = (&)

3. [ wuv ]=[uvw ]=[ vwu ]

$. (u x v) = u. (v x $) = v. ($ x u)

$. (u x v) = ($1! $2! $3) . [(u2v3 u3v2)! (u3v1 u1v3)! (u1v2

u2v1)"= u2$1v3 u3$1v2# u3v1$2 u1$2v3# u1v2$3

u2v1$3

u. (v x $) = (u1! u2! u3) . [(v2$3 v3$2)! (v3$1 v1$3)!

(v1$2 v2$1)"= u2$1v3 u3$1v2# u3v1$2 u1$2v3# u1v2$3

u2v1$3

'. u ( v w )=(u . v ) w+(u . w ) v

u ( v w )

u |v1 v2 v3w1

w2

w3|


3/9

u [ ( v2 w3v3 w2 ) , ( v3 w1v1 w3 ) , (v1 w2v2 w1 ) ]

| u1 u2 u3v2

w3v

3w

2 v

3w

1v

1w

3 v

1w

2v

2w

1|

=

(u2 v1 w2u2 v2w 1u3 v3 w1+u3 v1 w3 ) , (u3v2 w 3u3 v3 w2u1 v1w 2+u1 v2 w1 ) ,(u1 v3w 1u1 v1 w3

(u . v ) w+ (u . w ) v

(u1 v1+u2 v2+u3 v3 )( w1 , w2 , w3 )+( u1 w1+u2 w2+u3 w3 ) ( v1, v2 , v3 )

(u1 v1w 1u2 v2 w1u3 v3 w1 ) , (u1 v1 w2u2 v2 w2u3 v3 w 2 ) , (u1 v1 w3u2 v2 w 3u3 v3 w 3)+

=

(u2 v1 w2u2 v2w 1u3 v3 w1+u3 v1 w3 ) , (u3v2 w 3u3 v3 w2u1 v1w 2+u1 v2 w1 ) ,(u1 v3w 1u1 v1 w3

5. u v2

# (u . v )2

= u2v2

u2v2=u

1

2v

1

2+u1

2v

2

2+u1

2v

3

2+u2

2v

1

2+u2

2v

2

2+u2

2v

3

2+u3

2v

1

2+u3

2v

2

2+u3

2v

3

2

(1)

(u . v )2=u

1

2v1

2+u2

2v2

2+u3

2v3

2+2u1v1u2v2+2u

1v1u3v3+2u

3v3u

2v2 (2)

u v2

=|u1 u2 u3v

1 v

2 v

3|2

=(u2 v3u3 v2 )2+(u3 v1u1 v3 )2+(u1 v2u2 v1 )2=u22 v32+u32 v222 u2 v3 u3 v2+

(3)

De (3) # (2)

u1

2v

1

2+u1

2v

2

2+u1

2v

3

2+u2

2v

1

2+u2

2v

2

2+u2

2v

3

2+u3

2v

1

2+u3

2v

2

2+u3

2v

3

2=u2v2


4/9

u v2 # (u . v )2

= u2v2

. [uv ( w+3 u+4 v )]= [uvw ]

[u . ( v (w+3u+4 v )) ]= [u ( v w+v 3u+v 4 v )]=u ( v w+3 u v )=u [ v w ]+u [3 u v ]

0

u [ v w ]=[ uvw ]

*. Determinar el volumen +el paralelep,pe+o +e aristas a = (1!

-2! 3)! = (1! 1! -2) y c = (2! -1! 1)! si existe.

/ = |[a b c]| = |a .(bxc )| = |(1,2, 3) .|

1 1 22 1 1||= |(1,2,3) .(1,5,3)| = |1+109| = 0 u3

. Sean a = (1! 2! 3)! = (2! ! ) y c tal ue c = 2! calcular

c tal ue c es ortogonal a a x y a # .

Sien+o

c=( c1 , c2, c3 )

a b=(1215,66,45)=(3,0,1 ) (c1 , c2 , c3 ). (3,0,1)=0

3 c1+c

3=0

a+b=(3,7,9) (c1 , c2 , c3 ). (3,7,9)=0 3 c1+7 c2+9 c3=0

c=26 c1

2+c22+c3

2=676

{ 3 c

1+c

3=0

3 c1+7 c

2+9 c

3=0

c1

2+c2

2+c3

2=676}


5/9

c3=3 c

1 c

1=

c3

3

c3+7 c

2+9 c

3=0 7 c

2=10 c

3 c

2=10 c

3

7

c32

9+

100

49c

3

2+c3

2=676 49 c32+ 00 c3

2+441 c32=298116

1390 c32=298116 c3

2=298116

1390 c3=2981161390

c2=

87

298116

1390 c1=

298116

13903

. Sea el vector x tal ue es ortogonal al e4e 5 y al vector a = (!

-2! 3). Si x 6orma un 7ngulo agu+o con el e4e 8 y llxll = 117 !

9allar x.

Sien+o x=(x1 , x2 , x3 )

X=(1,0,0)

(x1 , x2, x3 ) . (1,0,0)=0 x1=0

(! -2! 3). (x1, x2, x3 )=0 5x12x2+3x3=0

x1

2+x22+x3

2=117

{ x

1=0

5x12x

2+3x

3=0

x1

2+x2

2+x3

2=117}2x

2+3x

3=0


6/9

x3=2

3x

2

02+x22+ 4

9x2

2=117 9x22+4x2

2=1053 13x22=1053 x2

2=81

x2= 9 x

3= 6 x

1=0

xA=(0,9,6) xB=(0,9,6)

Sien+o % el 7ngulo entre x y el e4e 8.

cos= x

. Z

xZ

cosA=6

117

C o C

cosB= 6

117

C o C

-10 - - -' -2 0 2 ' 10

-10

-

0

10

Y

x=(0,9,6)

10. alcular (a # ).[(a x c) x (a # ) # c" ! +on+e a = (1! 1! 1)!

= (2! 1! 3) y

c = (1! 1! ').


7/9

a+b=(3,2,4)

a c=|1 1 11 1 4|=(3,3,0 )

(a c ) (a+b )=|3 3 03 2 4|=(12,12,15)

(a+b ) . [(a c ) ( a+b )+c ] (3,2,4) . [(12,12,15 )+ (1,1,4 )]

(3,2,4) . (11,11,19 ) 3322+76=21

11. Sean los vectores unitarios a! ! c tal ue y c sonortogonales. alcular

(a x ).(a x ) (c x ).( x c) # (a.)(a.) .

a=b=c=1

b . c=0

Por propiedad:ab2=ab(a.b )2

(a b ) . (a b )+(b c ) . (b c )+(a . b ) (a . b ) ab2+bc2+(a .b )2

a1

b1

(a .b )2+b1

c1

sin1

+(a .b )21+1=2

12. ;alle el volumen +el tetrae+ro 6orma+o por los siguientes

vectoresa = (1! -2! ')! = (3! -! 3) y c = (3! -2! 1)! si existe.


8/9

/ = |[a b c ]6 | = |(1,2, 4).|3 5 33 2 1|6

| = |(1,2, 4).(1,6,9)6 | =|112+36|

6 =25

6 u3

13. Si el volumen +e un tetrae+ro regular es


9/9

1. ;alle la ecuacin +el plano si su vector normal al plano est7

conteni+o en el e4e A y corta al segmento por la tercera

parte +on+e

Documents

Pract. Dirigida #5-6