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 !"#$%&'&(!# )( *!"+! A = (x1, y1, z1) & )( ,&+!' "-! ").!
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#!(&"+& #&8 ! ,&+!'
−→
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−→
AP = t~v :;/<=
*2'2 2.>)( t ∈ R/
? *2'+$' %2 &3)24-! :;/<=8 ,&'$fi62(!# 3)&
P −A = t~v
!) 2$"%2
P = A + t~v, :;/A=
3)& &( 6!!'%&"2%2# fi62
(x, y, z) = (x1, y1, z1) + t(a, b, c) :;/B=
C)2.3)&' )(2 %2# &3)24D&# :;/<=8:;/A= !) :;/B= 9 %&"!($"2%2
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%2 '&+2 r & t 9 %&"!($"2%! ! 8)$9'!"$#/
:;!'8,# <=>=2= ? '&+2 r 3)& *2##2 *!' A = (1,−1, 4) &
+&( 2 %$'&4-! %& ~v = (2, 3, 2) +&( &3)24-! ,&+!'$2. %& 26!'%!
6!( :;/B=F
r : (x, y, z) = (1,−1, 4) + t(2, 3, 2)
&( 3)& (x, y, z) '&*'&#&"+2 )( *!"+! %& r 2'G$+'H'$!/ I2'2
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 ! "#$%& ! &%'(!)
t = 1 ⇒ P1 = (1,−1, 4) + 1 · (2, 3, 2) = (2, 3, 6)
t = 0 ⇒ P0 = (1,−1, 4)
t = −1 ⇒ P
−1 = (−1,−4, 2)
t = 3 ⇒ P3 = (7, 8, 10)
*(+,&' -)./0 P = A + t~v)
 !"#$%&'() 1* 2 %3,'45 3,% &%6&%!%"7' ' &%7' r " %8%$69 
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6 (! B'!7' 7 $'& ,7& 6 "7 ?% r %$ >%C ? 6 "7 A= ,
 ,7& >%7 & 3,'93,%& "5 ",9 3,% !%D' $#97(69 ?% ~v= 6 &
%8%$69 =
(x, y, z) = (1,−1, 4) + t(4, 6, 4)
: ,7&' %3,'45 >%7 &('9 ?% r %$ 3,% !% ,7(9(C , >%7 & 2~v =
(4, 6, 4) ; $ >%7 & ?(&%7 & %$ >%C ?% ~v = (2, 3, 2))
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 ! "#$!%&' (")'*+!, -! *")!
(x, y, z) = (x1, y1, z1) + t(a, b, c)
'$ !+.-!
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
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z = z1 + ct
89:;<
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-! *")!:
9:!'7,# ;<=<2< = *")! r #$" /!77! /",' /'.)'A = (3,−4, 2)
" 4 /!*!,",! !' (")'* ~v = (2, 1,−3)2 -" !0'*-' 0'5 89:;<2 )"5
"#$!%>"7 /!*!54)*+0!7
r :
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
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z = 2− 3t
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= *")! -"fi.+-! /",'7 /'.)'7 A " B 4 ! *")! #$" /!77! /'* A
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−−→
AB:
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#$" /!77! /'* A = (3,−1,−2) " B = (1, 2, 4):
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−−→
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!,'("!-
r :
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
x = 3− 2t
y = −1 + 3t
z = −2 + 6t
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( $(># ! -(9"($'! AB ;!*/9(" A ( (='*("/%#%( BB: C- (10#6
2D(- )('!*/#/- %!- -(9"($'!- AB ( BA 5!" 0 ≤ t ≤ 1 -3!
P = A + t(B −A) ( ;?:AB
P = B + t(A−B), ;?:EB
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t =
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a
t =
y − y1
b
t =
z − z1
c
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x− 3
2
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y
2
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z + 5
−1
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y
2
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x− x1
a
=
y − y1
b
"
x− x1
a
=
z − z1
c
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a
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a
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!001/ >':0.!.!/'0 #$" y " z 2'3"/ 0"6 3! 0"?$1:." 4'6/!@
y = mx + n " z = px + q.
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P = (x,mx + n, px + q)- "/ #$"
m =
b
a
" n = y1 − b
a
x1
p =
c
a
" q = z1 − c
a
x1
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"#$!%D"0 01/E.61>!0@
x− 2
1
=
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2
=
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−3
F !001/- 4!5":3'
x− 2
1
=
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2
"
x− 2
1
=
z + 3
−3
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*&." !/!" &0!+!-!( $ P = (x, 2x − 8,−3x + 3)) ∀x ∈ R) !(
12! P 3 2( *&,#& $'0.#'4'.& ,$ '!#$ r5
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+! r "?&G 

x = −1 + 2t
y = 2 + 3t
z = 4
H$" ,& -$"& +$ fiD2'$ 9E5FE:) $" !12$78!" *$'$(3#'.-$" +! r
"?&G 

x = 1 −t
y = 5
z = 2t
83
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/0 1#/ A = (−1, 2, 4) / ~v =
(2, 3, 0) 2~v//*+%,-− xy'
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(1, 5, 0) / ~v = (−1, 0, 2)'
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40% $/5% 6 *%$%+/+% % #0 7-3 /!8-3 2eixo − x. eixo − y
-# eixo− z9 3/ 3/#3 :/5-$/3 7!$/5-$/3 ;-$/0 *%$%+/+-3 % ~i =
(1, 0, 0) -# % ~j = (0, 1, 0) -# % ~k = (0, 0, 1)' </35/ =%3-. %&$'
%$' ()*+),",#"' %) -"#). '/) ,&0$''
12"*+0) 345464 >/?% r % $/5% 1#/ *%33% *-$ A = (2, 3, 4) /
5/0 % 7!$/@A- 7- :/5-$ ~v = (0, 0, 3)' B-0- % 7!$/@A- 7/ ~v 6 %
0/30% 7/
~k. *-!3 ~v = 3~k. % $/5% r 6 *%$%+/+% %- /!8- eixo−z'
C $/5% r *-7/ 3/$ $/*$/3/,5%7% */+%3 /1#%@D/3

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x = 2
y = 3
z = 4 + 3t
C3 fi"#$%3 2&'(F9 / 2&'(G9 %*$/3/,5%0 $/5%3 1#/ *%33%0 *-$
A = (x1, y1, z1) / 3A- *%$%+/+%3 %-3 /!8-3 eixo− y / eixo− x.
84
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5
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$+,-+./!0%1+2/+' 3#%, +4#%56+, ,789 :+ ;8$1% $+,-+./!0%9


x = x1
y = y1 + k · t
z = z1
+


x = x1 + k · t
y = y1
z = z1,
.81 k ∈ R fi=8 + t -%$>1+/$8'
 !"#$% &'(?* A = (x1, y1, z1) +
~v = ~j'
 !"#$% &'(@* A = (x1, y1, z1) +
~v = ~k'
 !" #$%& $'()*$& +,-+,%./$/.&
3!fi+(56# 7899 A !"#$%& '( )#*& +(,*&-8 3+B%1 %,
$+/%, r1 + r2 .81 %, :!$+56+, :+ ~v1 + ~v29 $+,-+./!0%1+2/+'
85
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 !"#"$%& %&'()* +" +($, -"#$, r1 & r2 ' #&(') *(+,-' .&
,# /&0') .1)&0') .& r1 & .& ,# /&0') .1)&0') .& r22 3&(.' θ
&%0& *(+,-'4 &(05'
cos θ =
|〈~v1, ~v2〉|
|~v1| |~v2| 6'# 0 ≤ θ ≤ .
π
2
7829:
./"01)* 234353 "-6,-") ' *(+,-' &(0)& "% )&0"%
r1 :


x = 3 + t
y = t
z = −1− 2t
& r2 :
x + 2
−2 =
y − 3
1
=
z
1
;&)6&<" =,& '% /&0')&% .1)&0')&% .& r1 & r2 %5'4 )&%>&601/"$
#&(0&4 ~v1 = (1, 1 − 2) & ~v2 = (−2, 1, 1)2 ?" &=,"@5' 7829:
>'.&#'% .&>)&&(.&) =,&
cos θ =
|〈~v1, ~v2〉|
|~v1| |~v2| =
|〈(1, 1,−2), (−2, 1, 1)〉|
|(1, 1,−2)| |(−2, 1, 1)| =
| − 2 + 1− 2|√
6
√
6
=
1
2
;')0"(0'4 θ = arc cos
(
1
2
)
=
π
3
)". = 60o2
6"fi&89:* 23;< 7 !"#$ %&"'(')#*$+3 3&A"# "% )&0"% r1
& r2 6'# "% .1)&@B&% .& ~v1 & ~v24 )&%>&601/"#&(0&2 C(05'
r1 ⊥ r2 ⇔ 〈~v1, ~v2〉 = 0
?,"% )&0"% ')0'+'("1% >'.&# %&) 6'(6'))&(0&% ', (5'2 D'
&(0"(0'4 ">&%") .& "#<"% "% )&0"% r1 & r2 ." fi+,)" 782FG:
%&)&# ')0'+'("1% " r4 (5' %5' 6'(6'))&(0&%4 " r & %1# >&)$
>&(.16,-")&%2
./"01)* 2343;3 ;')0"(0'4 "% )&0"% r1 & r2 ."."% " %&+,1)
86
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01/ /$2/"/3%!0'


x = t
y = −2t + 1
z = 4t
,


x = 3− 2t
y = 4 + t
z = t
4/!0 0,35/ ~v1 = (1,−2, 4) , ~v2 = (−2, 1, 1) 6,2/$,0 5!$,2/$,0
5, r1 , r2 ,
〈~v1, ~v2〉 = 1(−2)− 2(1) + 4(1) = 0,
%0 $,2%0 r1 , r2 01/ /$2/"/3%!0'
3!fi+(56# 789: 7 !"#$ %&"%'%(#)$ # *+#$ &!"#$,8 8,9
:%- %0 $,2%0 r1 , r2 31/ ;%$%<,<%0+ =/- %0 5!$,>?,0 5, ~v1 ,
~v2+ $,0;,=2!6%-,32,+ ,321/ 2/5% $,2% r 0!-#<2%3,%-,32, /$9
2/"/3%< % r1 , r2 2,$@ % 5!$,>1/ 5, #- 6,2/$ ~v+ 2%< A#,


〈~v,~v1〉 = 0
〈~v,~v2〉 = 0
7&'BC
87
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'( 345678 1!'(,9)+!& *&),
~v = ~v1 × ~v2 345:7
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1!#2!&5
%&"'()* +,-,., =(2(,+"#), (>*)/?(& 1),)+%2,"-)& ') ,(2)
r >*( 1)&&) 1(.! 1!#2! A = (3, 4,−1) ( % !,2!@!#). A& ,(2)&
r1 : (x, y, z) = (0, 0, 1)+t(2, 3,−4) ( r2 :


x = 5
y = t
z = 1− t.
 & '",(/?(& '( r1 ( r2 &0! '(fi#"')& 1(.!& $(2!,(& ~v1 = (2, 3,−4)
( ~v2 = (0, 1,−1)5 &&"+8
~v1 × ~v2 =
∣∣∣∣∣∣∣∣∣
~i ~j ~k
2 3 −4
0 1 −1
∣∣∣∣∣∣∣∣∣
= (1, 2, 2)
C!,2)#2!8 r :


x = 3 + t
y = 4 + 2t
z = −1 + 2t
%&"'()* +,-,/ 3 !"#$%#&'( )# *+,% -#",%., D)+!&
$(,"fi-), &( )& ,(2)& r1 ( r2 &0! -!#-!,,(#2(& (8 (+ -)&! )fi,E
+)2"$!8 '(2(,+"#), ! 1!#2! '( "#2(,&(/0!F
3)7
r1 :


x = 3 + h
y = 1 + 2h
z = 2− h
( r2 :


x = 5 + 3t
y = −3− 2t
z = 4 + t
88
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
5
 !" 
 !"
r1 :


y = x
z = 1− x
# r2 :


x = −t
y = 1 + t
z = 2t
 $"
r1 :


y = x + 2
z = −x− 1
# r2 :
x + 1
−2 =
y − 1
−2 =
z + 1
2
.
%# #&'()'* +, -./). (x, y, z) $.,+, 0( 1+2( *#)2(3 (+2( $.4
.*1#/212( .!#1#$#, 2 ).12( 2( #5+267#( 1# r1 # r28
3#,456#/ 2" 9:+2;2/1. 2( #&-*#((7#(3 )#,.( 5+#


3 + h = 5 + 3t
1 + 2h = −3− 2t
2− h = 4 + t
⇒


h− 3t = 2
2h + 2t = −4
−h− t = 2
<.*)2/).3 2 (.;+6=. > h = t = −1 #3 2((',3 (2,−1, 3) .
-./). 1# '/)#*(#6=. #/)*# 2( *#)2( r1 # r28
3#,456#/ !" ?2@#/1. 2( 1#A'12( (+!()')+'67#(3 )#,.( . ('()#,2


1 + t = −t
2t = 1 + t
B -2*)'* 1'((.3 $./()2)2,.( 5+# t = − 1
2
# t = 18 <.*4
)2/).3 $.,. . ('()#,2 /=. )#, (.;+6=.3 /=. #&'()# +,
-./). 1# '/)#*(#6=.8
3#,456#/ $" C!(#*A# 5+# .( A#).*#( 1'*#).*#( 1# r1 # r2 (=.3 *#(-#$4
)'A2,#/)#3 ~v1 = (1, 1,−1) # ~v2 = (−2,−2, 2)3 .+ (#D23
~v2 = −2~v18 <.*)2/).3 2( *#)2( (=. -2*2;#;2( # /=. $.'/$'4
1#/)#(3 -.'( . -./). (0, 2,−1) ∈ r13 ,2( (0, 2,−1) /∈ r28
89
 !"#$
 !""#$% &'( )*#"+) ,$ -(&+( .) #&+)/")0'( )&+/) !"
/)+!" r1 ) r21
 !" #$%&'(
2)"+! !,3!% .)fi&#$(" ! )5,!0'( 6)+(/#!3 .! /)+! )% -!/! #""(%
,"!$(" !-)&!" ,$ 6)+(/ ) ,$ -(&+( .( -3!&( 7(, .( )"-!0(8
-!/! .)fi&#93!1 :(&;)<)$(" (,+/! =(/$! .) /)-/)")&+>93!% #"+(
?% !+/!6?" .) ",! )5,!0'( -!/!$?+/#<!% .)"</#+! -(/ !3@,$!"
)5,!0A)" 5,) .)-)&.)$ .) !-)&!" ,$ -!/B$)+/(1 :($ C!")
&! .)fi&#0'( .! )5,!0'( 6)+(/#!3 .! /)+!% .)fi&#$(" +!$C?$
,$! /)+! -(/ .(#" -(&+(" ) ,$ ")@$)&+( -!/!$)+/#D!.(1 :(9
&;)<)$(" -/(-/#).!.)" #$-(/+!&+)" .!" /)+!"% <($( ( -!/!9
3)3#"$( .) /)+!" /)3!+#6( !(" -3!&(" ) )#*(" <((/.)&!.("% )
B&@,3(" )&+/) .,!" /)+!"1
 !)* +,-.-/0/$%
7!8 E)+)/$#&) ,$! )5,!0'( 6)+(/#!3 .! /)+! r .)fi&#.! -)3("
-(&+(" A = (2,−3, 4) ) B = (1,−1, 2) ) 6)/#fi5,) ") ("
-(&+(" C = (
5
2
,−4, 5) ) D = (−1, 3, 4) -)/+)&<)$ F r1
7C8 E!.! ! /)+! r : (x, y, z) = (−1, 2, 3)+t(2,−3, 0)% )"</)6!
)5,!0A)" -!/!$?+/#<!" .) r1
7<8 E)+)/$#&) !" )5,!0A)" -!/!$?+/#<!" .! /)+! 5,) -!""!
-)3(" -(&+(" A ) B &(" ")@,#&+)" <!"("G
#1 A = (1,−1, 2) ) B = (2, 1, 0)H
##1 A = (0, 0, 0) ) B = (0, 1, 0)1
90
 !"#$!% ! &!#'!"$() *+),-"(.)/ 0(1$# 2
5
 !" 
 !" # $%&'% P = (m, 1, n) $()'(&*( + )(', -.( $,//, $%)
A = (3,−1, 4) ( B = (4,−3,−1)0 1('(23&( P 0
 (" 4()3fi-.( /( %/ $%&'%/ P1 = (5,−5, 6) ( P2 = (4,−1, 12)
$()'(&*(2 + )(',
r :
x− 3
−1 =
y + 1
2
=
z − 2
−2 .
 6" 1('()23&( % $%&'% !, )(', r :
x− 1
2
=
y + 3
−1 =
z
4
-.(
'(27
30 ,8/*3//, 9:
330 %)!(&,!, ;0
 <" 1('()23&( % =&<.>% (&')( ,/ )(',/7
r1 :


x −2− t
y = t
z = 3− 2t
( r2 :
x
2
=
y + 6
1
=
z − 1
1
 ?" 1('()23&( % @,>%) !( n $,), -.( /(A, !( 30o % =&<.>%
(&')( ,/ )(',/
r1 :


y = nx + 5
z = 2x− 2
( r2 :
x− 2
4
=
y
5
=
z
3
 3" 4()3fi-.( /( ,/ )(',/ , /(<.3) /B% *%&*%))(&'(/ (C (2 *,/%
,fi)2,'3@%C (&*%&')( % $%&'% !( 3&'()/(DB%7
r1 :


x = 2− t
y = 4− t
z = −t
( r2 :


x = −3 + 6h
y = 1 + 7h
z = −1 + 13h
91