Question

¿Alguien que sepa desarrollar estos ejercicios sobre la unidad de vectores, en geometría y trigonometría?

Answer 1

Al efectuar las operaciones se obtiene:

1. (26, 46)

2. u = (√34, 30.96°); y = (8, 70°); v = (√85, 77.47°); w = (5√2, 81.87°)

3. x = (4.69, 1.71) ; y = (2.73, 7.51)

4. u·v = 37

5. x·y = 25.71

6. uₙ = (5, 3)/√34 ; vₙ = (2, 9)/√85

7. Θ_uv = 46.51° ; Θ_vw = 4.4° ; Θ_xy = 50°

8.  u×v = 39k ; w×y = -11.6k

9. 5u+4x-7y = (51.65, 30.73)

10. 9u+8w-12y = (20.24, -7.12)

Explicación:

Datos;

u = (5, 3)

v = (2, 9)

w = (1, 7)

x = (5, 20°)

y = (8, 70°)

1. 3u - 5w + 8v

3u = 3(5, 3) = (15,9)

5w = 5(1, 7) = (5, 35)

8v = 8(2, 9) = (16, 72)

Sustituir;

3u - 5w + 8v = (15,9) - (5, 35) + (16, 72)

3u - 5w + 8v = (15-5+16, 9-35+72)

3u - 5w + 8v = (26, 46)

2. Forma polar u, v, y, w;

(modulo, ángulo) = (√[(x)²+(y)²] , Θ = tan⁻¹(y/x))

|u| =√[(5)²+(3)²] = √34

Θ = tan⁻¹(3/5) = 30.96°

u = (√34, 30.96°)

|v| =√[(2)²+(9)²] = √85

Θ = tan⁻¹(9/2) = 77.47°

v = (√85, 77.47°)

|w| =√[(1)²+(7)²] = 5√2

Θ = tan⁻¹(7) = 81.87°

w = (5√2, 81.87°)

3. Forma cartesiana (rcos(Θ), rsen(Θ));

x = (5, 20°) = (5cos(20°), 5sen(20°)) = (4.69, 1.71)

y = (8, 70°) = (8cos(70°), 8sen(70°)) = (2.73, 7.51)

4. u·v

Producto escalar;

u·v = (5, 3) · (2, 9)

u·v = (5×2)+(3×9)

u·v = 10 + 27 = 37

5. x·y

Producto escalar;

x·y = (5cos(20°), 5sen(20°)) ·  (8cos(70°), 8sen(70°))

x·y = (5cos(20°)×8cos(70°))+(5sen(20°)×8sen(70°))

x·y = 25.71

6. Normalizar los vectores;

A = r/|r|

Sustituir;

uₙ = (5, 3)/√34

vₙ = (2, 9)/√85

7. Ángulos de separación;

Θ_uv = Θ_v - Θ_u

Sustituir;

Θ_uv = 77.47° -  30.96°

Θ_uv = 46.51°

Θ_vw = 81.87° - 77.47°

Θ_vw = 4.4°

Θ_xy = 70° - 20°

Θ_xy = 50°

8. u×v ; w×y

Producto vectorial;

$uxv=\left[\begin{array}{ccc}i&j&k\\5&3&0\\2&9&0\end{array}\right]$

u×v = (45-6)k

u×v = 39k

$wxy=\left[\begin{array}{ccc}i&j&k\\1&7&0\\2.73&7.51&0\end{array}\right]$

w×y = (7.51-19.11)k

w×y = -11.6k

9. 5u+4x-7y

= 5(5, 3)+ 4(4.69, 1.71) - 7(2.73, 7.51)

= (25+18.76-19.11, 15+6.84-52.57)

= (51.65, 30.73)

10. 9u+8w-12y

= 9(5, 3)+8(1, 7)-12(2.73, 7.51)

= (45+8-32.76, 27+56-90.12)

= (20.24, -7.12)