UNIDAD I.

ACTIVIDAD 2. “EJERCICIOS CON VECTORES”

03 / Febrero / 2016

1. Encuentra la suma de los vectores u+v para cada inciso:

a) u = (5, -3) , v = (4, 2) b) u = (1, 7) , v = (2, -2)

ra = u + v = (5, -3) + (4, 2) = (9, -1) rb = u + v = (1, 7) + (2, -2) = (3, 5)

ra = (9, -1) rb = (3, 5)

Suma de los tres vectores

c) u = (-11, -6), v = (13, 9) R.= r1 + r2 + r3 = (9, -1) + (3, 5) + (2, 3) =(14,7)

r3 = u + v = (-11, -6) + (13, 9) = (2, 3) R = (14,7)

r3 = (2, 3)

2. Encuentra la magnitud del vector resultante de la suma de los vectores anteriores.

ra = (9, -1) | | √ √ rb = (3, 5) | | √ √

| | √ = 9.0553 | | √ = 5.8309

rc = (2, 3) | | √ √ Magnitud de la resultante R

| | √ = 3.6055 R = (14, 7) | | √ √

R = √ = 15.6524

ACTIVIDAD 2. “EJERCICIOS CON VECTORES”

3. Representa la suma de vectores en el plano cartesiano.

a) u= (B) = (5, -3) , v=(C) = (4, 2) b) u= (B) = (1, 7) , v=(C) = (2, -2)

ra = u + v = (5, -3) + (4, 2) = (9, -1) rb = u + v = (1, 7) + (2, -2) = (3, 5)

(b) = ra = (9, -1) = 9i - j = (B´) (b) = rb = (3, 5) = 3i + 5j = (B´)

(9, -1)

(3, 5)

𝒂𝒏𝒈𝒖𝒍𝒐 𝐜𝐨𝐬−𝟏 𝟏𝟒

√𝟑𝟒 ∗ 𝟐√𝟓 𝟓𝟕.𝟓𝟐°

Componentes

Vector unitario =ra = ( 𝟗

√𝟖𝟐 ,

𝟏

√𝟖𝟐 )

Angulo entre Vectores

𝒂𝒏𝒈𝒖𝒍𝒐 𝐜𝐨𝐬−𝟏 𝟏𝟐

𝟓√𝟐 ∗ 𝟐√𝟐 𝟏𝟐𝟔.𝟖𝟔°

Componentes

Vector unitario =rb = ( 𝟑

√𝟑𝟒 ,

𝟓

√𝟑𝟒 )

Angulo entre Vectores

3. Representa la suma de vectores en el plano cartesiano.

Suma de los tres vectores

c) u = (-11, -6), v = (13, 9) R.= ra + rb + rc = (9, -1) + (3, 5) + (2, 3) =(14,7)

rc = u + v = (-11, -6) + (13, 9) = (2, 3) ( f ) = R = (14,7) = 14i + 7j =(B´)

( b ) = rc = (2, 3) = 2i + 3j = (B´)

(2, 3)

(14, 7)

𝒂𝒏𝒈𝒖𝒍𝒐 𝐜𝐨𝐬−𝟏 𝟏𝟗𝟕

√𝟏𝟓𝟕 ∗ 𝟓√𝟏𝟎 𝟔.𝟎𝟖°

Componentes

Vector unitario =rc = ( 𝟐

√𝟏𝟑 ,

𝟑

√𝟏𝟑 )

Angulo entre Vectores

𝒂𝒏𝒈𝒖𝒍𝒐 𝐜𝐨𝐬−𝟏 𝟑𝟗

√𝟖𝟒 ∗ √𝟑𝟒 ∗ √𝟏𝟑 𝟕𝟖.𝟑𝟐°

Componentes

Vector unitario =R = ( 𝟏𝟒

𝟕√𝟓 ,

𝟕

𝟕√𝟓 )

Angulo entre Vectores

4. Encuentra la resta de los siguientes vectores. Y represéntalos en el plano.

1. u = (1, 1, 2) , v = (0, 2, 1)

r1 = u - v = (1, 1, 2) - (0, 2, 1) = (1, -1, 1)

b= r1 = (1, -1, 1) = i – j + k =(B´)

𝜶 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒊 𝐜𝐨𝐬−𝟏𝟏

√𝟑 𝟓𝟒.𝟕𝟑°

𝜷 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒋 𝐜𝐨𝐬−𝟏 𝟏

√𝟑 𝟏𝟐𝟓.𝟐𝟔°

𝜸 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒌 𝐜𝐨𝐬−𝟏𝟏

√𝟑 𝟓𝟒.𝟕𝟑°

Componentes

Vector unitario =r1 = ( 𝟏

√𝟑 ,

𝟏

√𝟑,

𝟏

√𝟑 )

Magnitud del vector |r1|= √𝟑

Cosenos directores del vector r1

2. u = (6, 0, 2) , v = (3, 5, 1)

r2 = u - v = (6, 0, 2) - (3, 5, 1) = (3, -5, 1)

b = r2 = (3, -5, 1) = i – 5j + k = (B´)

|

𝜶 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒊 𝐜𝐨𝐬−𝟏𝟑

√𝟑𝟓 𝟓𝟗.𝟓𝟐°

𝜷 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒋 𝐜𝐨𝐬−𝟏 𝟓

√𝟑𝟓 𝟏𝟒𝟕.𝟔𝟖°

𝜸 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒌 𝐜𝐨𝐬−𝟏𝟏

√𝟑𝟓 𝟖𝟎.𝟐𝟔°

Componentes

Vector unitario =r2 = ( 𝟑

√𝟑𝟓 ,

𝟓

√𝟑𝟓,

𝟏

√𝟑𝟓 )

Magnitud del vector |r2|= √𝟑𝟓

Cosenos directores del vector r1

3. u = (6, 1) , v = (7, 1)

r3 = u - v = (6, 1) - (7, -1) = (-1, 2)

c = r3 = (-1, 2) = -i + 2j = (B´)

(-1, 2)

𝜶 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒊 𝐜𝐨𝐬−𝟏 𝟏

√𝟓 𝟏𝟏𝟔.𝟓𝟔°

𝜷 𝒂𝒏𝒈𝒖𝒍𝒐 𝒅𝒆 𝒋 𝐜𝐨𝐬−𝟏𝟐

√𝟓 𝟐𝟔.𝟓𝟔°

Componentes

Vector unitario =r3 = ( 𝟏

√𝟓 ,

𝟐

√𝟓 )

Magnitud del vector |r3|= √𝟓

Cosenos directores del vector r1