Answer:
a) <-1,5>, <1,-5>, <2,-10>, <-2,10>, <-100,500> are elements in set A
b) <1,4,0>, <0,0,0>, <2,3,0>, <18,32,0>, <8,7,0> are elements in set B
Step-by-step explanation:
a) Set A defines an infinite line in [tex]{\displaystyle \mathbb {R} } ^{2}[/tex], that means, 2-D space, that has a direction defined by the vector <-1,5>, and a parameter t that allows to get all the points in space in that direction both sides of the origin because t can be any real number.
We can get all the points of that line by plugging numbers into the parameter, that multiplies each component of the original vector, for example, plugging 1, -1, 2, -2 and 100, gives us the following points in space:
1: <-1,5>
-1: <1,-5>
2: <2,-10>
-2: <-2,10>
100: <-100,500>
b) Set B defines an infinite plane in [tex]{\displaystyle \mathbb {R} } ^{2}[/tex], that means, 3-D space, and it is defined by the vectors <2,-3,0> and <-1,1,0>, and parameters p and q, both being any independent from each other real number.
We can get all points in space belonging to that plane by plugging numbers into the parameters, for example, a: 1 and 1, b: 0 and 0, c: 1 and 0, d: 10 and 2, e: 3 and -2:
a: 1*<2,3,0>+1*<-1,1,0> = <1,4,0>
b: 0*<2,3,0>+0*<-1,1,0> = <0,0,0>
c: 1*<2,3,0>+0*<-1,1,0> = <2,3,0>
d: 10*<2,3,0>+2*<-1,1,0> = <18,32,0>
e: 3*<2,3,0>+(-2)*<-1,1,0> = <8,7,0>
