Question

1) Geometria Analítica ????

Answer 1

a) u = (1,-1)
||u|| = ||(1,-1)|| = √(1)² + (-1)²
       = √1 + 1 = √2

||u|| = √2

b) v = (-3,4)
||v|| = ||(-3,4)|| = √(-3)² + 4²
      = √9 + 16 = √25 = |5| = 5

||v|| = 5


c) w = (8,-6)
||w|| = ||(8,-6)|| = √(8)² + (-6)²
       = √64 + 36 = √100 = |10| = 10
||w|| = 10

d) ||u + v||
u = (1,-1)
v = (-3,4) 

||u + v|| = ||(1,-1) + (-3,4)|| = ||(1+(-3),-1 + 4)|| = ||(1-3, -1+4)|| = ||(-2, 3)||
||u + v|| = ||(-2,3)||

||(-2,3)|| = √(-2)² + 3² = √4 + 9 = √13

||u + v|| = ||(-2,3)|| = √13

e) ||2u - w||
u = (1,-1)
w = (8,-6)
||2u - w|| = ||2(1,-1) - (8,-6)|| = ||(2,-2) - (8,-6)|| = ||(2-8,-2-(-6))|| =
= ||(-6, -2+6)|| = ||(-6, 4)||

||(-6, 4)|| = √(-6)² + 4² = √36 + 16 = √52

||2u - w|| = ||(-6, 4)|| = √52

f) ||w - 3u||
w = (8,-6)
u = (1,-1)

||(8,-6) - 3(1,-1)|| = ||(8,-6) + (3,-3)|| = ||(8+3,-6-3)|| = ||(11,-9)||

||(11,-9)|| = √11² + (-9)² = √121 + 81 = √202

||w - 3u|| = ||(11,-9)|| = √202

g) v/||v||
v = (-3,4)
(-3,4)/||(-3,4)||

||(-3,4)|| = √(-3)² + 4² = √9 + 16 = √25 = 5

(-3,4)/5 = (-3/5,4/5)
v/||v|| = (-3/5,4/5)

h) ||(u/||u||)||
u = (1,-1)
((1,-1)/||(1,-1)||)

||(1,-1)|| = √(1)² + (-1)² = √1 + 1 = √2

(1,-1)/√2 = (1/√2,-1/√2)

||(1/√2,-1/√2)|| = √(1/√2)² + (-1/√2)² = √1/2 + 1/2 = √1 = 1

||(u/||u||)|| = 1