Algunos Ejercicios Resueltos Larson

1. u = (5, 12)
v = (-3, 2)
Calcular
a) u • v = (5 x -3) + (12 x 2) = -15 + 24 = 9
b) u • u = (5 x 5) + (12 x 12) = 25 + 144 = 169
c) ǁuǁ2 = (52+ 122)2 = (25+144)2 = (169)2= 132=169d) (u • v) v = 9 (-3,2) = (-27, 18)
e) u • (2v) = (5,12) • (-6, 4) = -30 + 48 = 18

2. u = (2, -3, 4)
v = (0, 6, 5)
Calcular
a) u • v = (2 x 0) + (-3 x 6) + (4 x 5) = 0 – 18 + 20 = 2b) u • u = (2 x 2) + (-3 x -3) + (4 x 4) = 4 + 9 + 16 = 29
c) ǁuǁ2 = (22+ (-3)2+ 42)2 = (4+9+ 16)2 = (29)2=29
d) (u • v) v = 2 (0, 6, 5) = (0, 12, 10)
e) u • (2v) = (2, -3, 4) •(0, 12, 10) = 0 - 36 + 40 = 4

3. u = 2î + ĵ – 2 = (2, 1, -2)
v = î - 3ĵ + 2 = (1, -3, 2)
Calcular
a) u • v = (2 x 1) + (1 x -3) + (-2 x 2) = 2 – 3 – 4 = -5
b) u • u = (2 x 2) + (1 x1) + (-2 x -2) = 4 + 1 + 4 = 9
c) ǁuǁ2 = (22+ 12+ (-2)2)2= (4+1+4)2 = 92=9
d) (u • v) v = -5 (1, -3, 2) = (-5, 15, -10)
e) u • (2v) = (2, 1, -2) • (2, -6, 4) = 4 – 6 – 8 = -10

4.ǁuǁ = 8, ǁvǁ = 5 y el ángulo entre u y v es π3. Calcular u • v
u • v = ǁuǁ ǁvǁ cos (60°) = 8 x 5 x ½ = 20

5. ǁuǁ = 40, ǁvǁ = 25 y el ángulo entre u y v es 5π6. Calcular u • v
u • v = ǁuǁ ǁvǁcos(150°) = 40 x 25 x cos(150 °) = -866

6. u = (1, 1), v = (2, -2). Hallar el ángulo Ɵ entre los vectores.
cos(Ɵ) = u • vǁuǁ ǁvǁ = 2-22 8 = 04 = 0
Ɵ = Arcos (0) = 90°

7. u = (3, 1), v =(2, -1). Hallar el ángulo Ɵ entre los vectores.
cos(Ɵ) = u • vǁuǁ ǁvǁ = 6-110 5 = 552 = 22
Ɵ = Arcos (22) = 45°

8. u = -2î + 3ĵ –
v = 2î + ĵ –
Hallar si u y v son ortogonales, paralelos oninguna de ambas cosas.
u • v = (-2 x 2) + (3 x 1) + (-1 x -1) = -4 + 3 + 1 = -4 + 4 = 0
Los vectores son ortogonales

9. u = (2, 3), v = (5, 1).
a) Proyectar u sobre v
proyvu = u • vǁvǁ2 v =10 + 325 + 1 (5, 1) = 12 (5, 1) = (52, 12)

b) Calcular el vector componente de u ortogonal a v
u = (2, 3) = w1 + w2
w1 = proyvu = (52, 12)

w2 = u – w1 = (2, 3) – (52, 12) = (-12, 52)...