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Física Volumen I
Capítul
Magnitud Vectorial: Son aquellas que para definirse, además Defnición gráfca de de un módulo y una unidad de medida, requieren de una dirección y un sentido.
A
Vector: Es un ente matemático que sirve para representar una magnitud vectorial.
Se lee: vector A
Partes de un vector
A
A
onde! Master your semester with Scribd "! #ódulo #agnitud, indica el tama$o del vector. & The New oYork Times ! irección, el $4.99/month. ángulo de inclinación. Special θ offer for students:es Only
! Sentido, indica indica %acia donde se dirige el vector
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Física Volumen I 2.
Capítul
Vector ctores es Colin olinea eale less:
colineales, se puede observar que todos los vectores prese En los vectores colineales, se
*odos aquellos vectores que se encuentran so(re una misma línea de acción. 3.
Vector ctores es Copl oplanal anales es::
*odos *odos aquellos vectores que se %allen en un mismo plano. 4.
Vectores Par Paralelos:
*odos aquellos que tengan la misma dirección, sin importar el módulo ni el sentido. 5.
Vectores Op Opuestos: os:
opuestos, siempre que se cumpla la siguiente igualda !os vectores serán opuestos,
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Física Volumen I #.
Capítul
Vectores $nitarios:
Se dice que por cada vector que e"iste también e"iste un vector unitario *odos aquellos que su módulo sea igual a .
A
$u
A
A A & A
%u %principales vectores unitarios
) vectores +os principales ( unitarios se traan en cada uno de los ejes c
* i j
'
j
(
'
i
&esco'posicin Vectorial:
Esta descomposición vectorial es llamada descomposición r A
You're Reading a Preview
A
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A"
-ara nunca conundirse, tener en cuenta que del eje en el cual nace el ángulo onde!
" / "cosθ "y / "senθ
"demás!
A
/ "i 0 "y 1
Master your semester with Scribd " / " 0 " & The New York Times
A A"
y
Special offer for students: tanθ Only / $4.99/month.
A
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Física Volumen I
Capítul
3ara este caso, solo nos fi1aremos en el sentido de estos, de tener el mismo sentido se sumaran, de tener sentido contrario se restarán.
*. Vectores con un +ngulo entre si:
Si dos vectores orman 34, su resultante será má"ima de ormar %534 será m todo del paralelogramo! 0ma" 6 0 6 0min /
A
A
/ #atemáticamente, %allaremos el módulo de la resultante con la siguiente epresión!
5/
"- 0 6- 0 -"6cosθ
%,lgunos Casos particulares:
0 & a7 1b7
a
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0&a 7
a
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b
a
0&a $
a
0&a
a
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7
938
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0& Not useful ;*
Useful
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Física Volumen I
Capítul
&i-erencia de dos Vectores:
Si:
a. Vectores colineales o paralelos:
A /
"l ser los vectores colineales o paralelos, solo nos fi1aremos en el sentido de estos, y de tener el sentido contrario se sumaran, de ser el mismo sentido se restaran.
!&A2/ A /
*. Vectores con un +ngulo entre si:
!&A1/
todo del paralelogramo!
/
A=/
Si dos vectores orman 34, su dierencia será mínima de ormar %534 será m !ma" 6 ! 6 !min You're Reading a Preview
A
Unlock full access with a free trial.
:ráficamente se puede compro(ar la diferencia por persecución de vectores. Download With Free Trial 3ara %allar el módulo de la resultante procederemos con la ley de cosenos o teorema Carnot.
"; 6 /
"- 0 6- ; -"6cosθ
%Casos particulares:
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a
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Física Volumen I
Capítul
u'a de '+s de dos vectores: 1. M"todo del Polgono (M"todo /r+-ico):
ados varios vectores, procedemos a ponerlos en persecución, uno a continuación de otro, el vector resultante será aquel que se trace desde el inicio del primer vector %asta el final del )ltimo vector.
En el primer esquema vemos que!se orma un polígono abierto, en consecuen
/
A
? ?
/
A
!
0
You're Reading a Preview Unlock full access with a free trial.
R= A + B + C + D
Download With Free Trial
! A
En el segundo esquema ? vemos que E se orma un polígono abierto, por lo que n
/
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Física Volumen I
.
Capítul
A = A x i + A y j + A z k B = B x i+ B y j + B z k
.
Entonces la resultante de
A
y
B
, será!
A
⃗ (¿ ¿ x + B x ) i + ( A y + B y ) j⃗ +( A z + B z )k R =¿ ⃗
.
⃗
3. M"todo por desco'posicin vectorial: >
A
/ @bicamos todos los vectores desde un mismo origen en un sistema de
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->
/ A A / !escomponemos cada vector en los ejes correspondientes tal como se vio a
A"
/" ?
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Sumamos como vectores colineales, hallando una resultante
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Física Volumen I
Capítul
No se descompone “C”, debido a que se halla sobre un eje coordenado. Productos:
-ara multiplicar un escalar por un vector, aplicaremos la propiedad distribu 1. $n 0scalar con un Vector:
A = A x i + A y j
ado el vector El escalar
@mA
Entonces!
m
A = m A x i + m A y j
mA
Si:
m&7
2. $n Vector por otro Vector: a. Producto 0scalar: Conocido como producto punto.
&
Si nos dan los módulos y el ángulo que forman, procederemos Este producto me dará como resultado un escalar, de al de la siguiente manera! Será má"imo siYou're 34 a Preview &Reading >34 Unlock full access with a free trial. A ∙ B = A B cosnulo θ si &
|
⃗
&
| | || |
⃗
3ara vectores unitarios!
i ∙ i =1
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i ∙ j= 0
j semester ∙ j =1unitarios j ∙ k =me 0 Master your with ?uando los vectores son Scribd iguales, daráRead uno to de ser dierentes m Free Foron 30this Days Sign up vote title & The New York Useful Not useful k ∙ k =Times i ∙ k =0 1 Special offer for students: Only $4.99/month.
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Física Volumen I &
Capítul
Si nos dan los módulos y el ángulo que forman, procederemos de la siguiente manera!
Este producto me dará como resultado un vector, el cual se dibuja perpendicu Será má"imo si & >34 | A X B|=| A||B|Senθ nulo si & 34 ⃗
⃗
En este producto no se cumple la propiedad conmutativa de la m &
Si!
A = A x i + A y j+ A z k
, además
B = B x i+ B y j + B z k &
Entonces!
|
|
You're Reading a Preview
⃗ j⃗ k Unlock full access with a free trial. ⃗ A X B A A A z = -ara calcular el producto cru, calcular la determinante de la m x solo ydebemos ⃗
&
i ⃗
B x
B y Download B z With Free Trial
3ara vectores unitarios!
Master your semester with Scribd Read Free Foron 30this Days Sign up to vote title i X i =0 j X j= 0 k X k =0 & The New York Times Useful Not useful Special offer for students: Only $4.99/month.
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Física Volumen I
Capítul )
a8 ?
(8
c8 D
d8
e8
". Calcular la resultante del sistema de vectores#
2 7"
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7"
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). %eterminar el mdulo de la resultante Unlock full access with a free trial. en cada caso#
9
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d+ 15
9
√ 3
938
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$;8
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Física Volumen I
Capítul d+ 5
√ 2 e+ 15 10
a+ b+ / c+ d+ - %98 e+ 5 7D 5. Sean los vectores 2 ( 5 3 4 ( cu3a
7
resultante es )
√ 10
ánulo entre ellos! a+ 1$0 b+ $ c+ )5
. ¿Cuál es el d+ 5$
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-. Se tienen dos vectores 2(/ 3 4(11. ¿Cuál de los siuientes no puede ser una resultante de ellos!
10. Se tienen dos vectores 2 mdulos 5 3 1. %eterminar el án 8orman los vectores si el vector 8orma un ánulo de / con el ma3or mdulo.
a+ )5 b+ -0 c+ $0 d+ 5$ 11. Se tienen dos vectores de md $. Calcular el ánulo entre los ve su resultante 8orma un ánulo el ma3or de ellos. a+ 1"
b+ 0
c+ 150
d+ -0
1". 6a resultante má&ima de dos es / 3 su resultante m7nima es ". valdrá su resultante si los vectore un ánulo de -0!
a+ ) b+ 5 c+ d+ 1$. Se tiene dos vectores que 8or a+ / b+ 11 c+ 1$ d+ " e+ 5 Unlock full access with a free trial. resultante má&ima iual a resultante m7nima iual a 1. . 6a resultante má&ima de dos vectores es Download With Free Trial resultante cuando dichos ve 1- 3 la m7nima es ). ¿Cuál es el mdulo de perpendiculares. la menor! You're Reading a Preview
a+ 10
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a+ "
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Física Volumen I
Capítul
1-. ara el sistema dado, encontrar una e&presin vectorial para & en 8uncin de a, b3c a+ c = b ; a " a b+ b = c ; a c+ b ; c ; a d+ b ; c = a c e+ = b ; c = a b 1. Ancontrar una e&presin vectorial para & en 8uncin de a, b 3 c a+ a = b = c 7b b+ = a ; b ; c c+ = a = b ; c d+ a ; b ; c a " 2b e+ a = b = c 1/. %eterminar una e&presin vectorial para & en 8uncin de a, b 3 c
c
b
"". Ancontrar la resultante de#
c
a
b
"$. %eterminar la resultante de#
√ 2 b a
√ 3
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d
c
a+ c = a ; b "). Ddenti8icar el valor de la result " Unlock full access with a free trial. b+ c ; a = b c+ c ; a ; b 2c Download With Free Trial d+ c = a = b a e+ = c ; a = b 1. %eterminar la resultante# a+ "b b b+ $a c+ "c c "5. 6a 8iura muestra un he&áon a d+ "a Master your semester with Scribd de lado ". Calcular el md Read Free Foron 30this Days Sign up to vote title e+ c resultante de los vectores mostrad & The New York Times Useful Not useful Cancel anytime. "0. %eterminar la resultante# Special offer for students: Only $4.99/month.
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Física Volumen I
Capítul
"-. Ebtener el vector & en 8uncin de los vectores 2 3 4. 1
a
A
Ancontrar el mdulo de la resultan vectores mostrados en cada caso. "/.
3
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Física Volumen I
Capítul d+ $
√ 2
resultante 8orme ;$ con positivo de las &. 2demás# 4("
e+ 5
(
√ 2
$$. 9allar 4 para obtener una resultante horiFontal a+ 1" $. %ados los vectores cu3os md b+ 11 73 7 D8 " 2("u, 4($u. Calcular el m c+ 15 d+ 1$ $8 vector . e+ "5 938 /
%9
$). Calcular el valor de α para que la resultante de los vectores mostrados sea vertical
c+
> 7
$7
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a+ 10 You're Reading a Preview
b+ $0 d+ @ u c+ )0 d+ 50 Withe+Free Trial u Download %98 e+ -0 7 $/. 9allar la manitud de la $5. Calcular el valor de θ para que la sistema de vectores que se mues 8iura. 6as manitudes de los resultante sea horiFontal.
D8
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$8 D8
$3
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Física Volumen I
Capítul
)$. 9alle el mdulo del vector r para los vectores establecidos. a+ 1
b+ "0
c+ $1
d+ 11
e+ /
)0. Calcular el mdulo del vector resultante del siuiente sistema de vectores.
a+
√ 2
b+
√ 5
c+ "
√ 3 √ 3
a+
√ 17
cm
d+ "
b+
√ 20
cm
e+
%cm
c+
√ 21 √ 13
cm
e+
– √ 3
cm
√ 2
√ 5 %38
%8
√ 3
√ 3
%cm
cm
d+
%
)). 6os mdulos de " vectore √ √ ❑ √ ❑ √ ❑ 2 √ 2 u halle el mdulo de la resultante 8orman entre s7 un ánulo de /". You're Reading a Preview
)1. %eterminar el mdulo de laUnlock suma 3 11 √ trial. full access with a+ a free di8erencia de vectores, si los mdulos respectivos son 1- 3 /. Download With Free Trial c+
b+
√ 29
5
√ 2 d+ 5
a+ "), ") b+ 1", / c+ /
√ 7
√ 3 , /
)5. %os 8uerFas cu3os mdulo
a
√ 3
938
√ 2
b
Master your semester with Scribd √ 7 , / d+ /New & The York Times
de /, determinar el mdulo de l Read Free Foron 30this Days Sign up to vote title resultante. Useful Not useful
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e+ /
√ 3
, /
3 10 8orman entre si u
a+
Cancel anytime.
√ 130
√ 130
b+
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Física Volumen I b+
√ 58
√ 58
c+
√ 53
√ 53
51. ¿GuH ánulo 8orma la resultan vectores dados con el eje 3!
d+
√ 57
√ 57
e+
√ 51
√ 51
). a+ 15 b+ "$ c+ 1 d+ "0 e+ 1$
d+ "5
a+ 1 b+ $ c+ 5 d+ e+
73 $8
√ 3
√ 5
5
9;8
73 $8
a+ /
√ 10
b+
√ 10
$8
%3
c+ 10
$;8
d+
5
D3
√ 10 √ 10
e+ N.2. 5$. 9allar el mdulo del vector sabe que la resultante de los Download With Free Trial mostrados es horiFontal b ( "5? c
a
a+ /0 b+ )0
√ 2
c+ /0
√ 2
7
Special offer for students: Only $4.99/month.
7%8
Unlock full access with a free trial.
$3
;8
D78
You're Reading a Preview
Master your semester with Scribd e+ "5 √ 7 & The New York Times 50.
%3
5". 9allar el mdulo del vector r *radio ( "5+
7D
)/. a+ *5,$+ b+ *5,"+ c+ *5,1+ d+ *",/+ e+ *$,@"+ ). a+ "5 b+ 5 c+ "5
Capítul
Read Free Foron 30this Days Sign up to vote title
d+ 1-0
Useful
e+ 100
√ 2
$8
D8
Not useful Cancel anytime.
c
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Física Volumen I
Capítul
55. An una semicircun8erencia con centro en “o” 3 radio “'”, se hallan contenidos ) vectores, halle el mdulo del vector suma. a+ ' b+ "' c+ $' d+ )' e+ cero
b+ " c+ $ d+ ) e+ 5 5. 9allar el vector & en 8unci vectores 2 3 4? sabiendo que I puntos medios de sus respectivos l
%738 o
A + B
5-. Iostramos los vectores espaciales, determine el valor del vector resultante en el cubo de lado 1.
a+ 1
b+
3
( A + B )
c+ d+
√ √ √ 2 22
B– A
√ 2
c+
√ 3
F "
a+ 1 b+
A
2
2
D8
/
A – B
√ 5
3 You're Readingd+ a Preview Unlock full access with a free trial.
e+ "
A B –
Download Withe+ Free Trial 3 6
5. An el diarama vectorial e&presar el mdulo del vector resultante. a+
√ 21
7 %
17 b+ √ Master your % semester with Scribd & The New York Times c+ √ 13
$
Special offer for students: Only $4.99/month.
√ 15
-0. 6a 8iura mostrada es un cuad pide e&presar : en 8uncin 2 3 4
a+
√ 2 ( A + B )
+B √ 2 ( A
Aon30this Read Free For Days Sign up to vote title
2
Not useful Cancel anytime.
Useful
√ 2 ( A – B )
'2
√ 2 ( A − B
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Física Volumen I
Capítul
-1. %eterminar en el cuadrado de la 8iura el vector & en 8uncin de 2 3 4
a+ *2 ; "4+
b+ *2 = "4+
c+ *2 ; 4+
√ 2 + 1
√ 2 + 1 "2
2
√ 3 /
b+ 10
√ 5
cm.
e+ 10 cm.
-). Jn vector horiFontal 8orma otro vector de 15unidades de determine el modulo de dicho vec manera que la resultante sea m7nim
2
A
2
a+ u d+ *2 = 4+
cm.
cm.
√ 5
d+ 5
√ 2 – 1 √ 2 – 1
√ 3
a+ 5
b+ 1"u
c+ 15u
d+ 1/u
√ 2 – 1 2
e+ N.2. -". 9allar el vector & en 8uncin de los You're Reading a Preview vectores 2 3 4 siendo la 8iura un cuadrado Unlock full access with a free trial.
A a+ ( √ 2 – 1 ) ( A + B ) b+
( √ 2+ 1 ) ( A + B )
c+
( √ 2 – 1 ) ( A – B )
'
-5. 8iura, determinar aG7 el modulo Download Withresultante. Free Trial
/
Master (your semester with Scribd ( ) ) + A – B 2 1 √ d+ & The New York Times Special e+ offerN.2. for students: Only $4.99/month.
-$
aG7 aG7
Si se sabe que el modulo de la
a+ 5 b+ 10 c+ 15 d+ "0 e+ "5
aG7
Read Free Foron 30this Days Sign up to vote title
Useful Notel useful --. %eterminar mdulo Cancel anytime.
resultante del conjunto mostrados en la 8iura.
de
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' A
Física Volumen I e+
√ 3
-. 9allar el mdulo del vector resultante, sabiendo que la 8iura es un cubo de arista “6”
Capítul a+ *"4 = 2+ b+ )*"4 = 2+1 c+ 1$*4 = "2+ d+ /*4 = "2+$5 e+ N.2.
a+ 6 b+ "6 c+ $6 d+ )6 e+ 56 -/. Al ánulo entre dos vectores de 5 3 10 unidades de lonitud cuando su resultante 8orma un ánulo de $ con el vector de ma3or mdulo es#
". Jna embarcacin parte del pu direccin sur@este recorriendo adentro, lueo vira /"L al norte recorriendo 150Om hasta llear puerto ¿GuH distancia ha3 entr puertos!
a+ $
$. An el triánulo 9allar el vec 8uncin de los vectores 2 3 4.
b+ )5
c+ -0
d+ 0
e+ 1"0
a+ )0Om d+ 1"0Om
b+ 0Om e+ 150Om
-. %eterminar la medida del ánulo α You're Reading a Preview a+ : ( */2 ; 54+/ para que la resultante de los vectores Unlock full access with b+ a:free(trial. *$2 = 54+/ mostrados sea iual a 10, sabiendo : ( *52 ; $4+/ además que 24(1", 4C(1- *I Download 3 N son Withc+Free Trial d+ : ( */2 ; 54+$ puntos medios+ e+ : ( *52 ; /b+$ a+ -0 b+ ) c+ 0 d+ 1"0 Master your e+ 1"
semester with Scribd & The New York Times
vectores 2? 4 3 C tienen Special 0. offer forKres students: Only $4.99/month. componentes “&” 3 “3” como se muestra
' A /
). 6a má&ima resultante de dos es "/u 3 su m7nima es )u. ¿Cu resultante cuando dichos vec ortoonales! Read Free Foron 30this Days Sign up to vote title a+ /u b+ 1"u c+ 1-u d+"0u
Useful
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Cuadro de F.T. Notab
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Física Volumen I => ?9>
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Capítul ? -9 -9 ?
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Download With Free Trial
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Read Free Foron 30this Days Sign up to vote title
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Useful
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