Chapter 4.2, Problem 42E

Rather than use the standard definitions of addition and scalar multiplication in $R^3$, let these two operations be defined as shown below.

(a) $\left(x_1,y_1,z_1\right)+\left(x_2,y_2,z_2\right)=\left(x_1+x_2,y_1+y_2,z_1+z_2\right)$

$c\left(x,y,z\right)=\left(cx,cy,0\right)$

(b) $\left(x_1,y_1,z_1\right)+\left(x_2,y_2,z_2\right)=\left(0,0,0\right)$

$c\left(x,y,z\right)=\left(cx,cy,cz\right)$

(c) $\left(x_1,y_1,z_1\right)+\left(x_2,y_2,z_2\right)=\left(x_1+x_2+1,y_1+y_2+1,z_1+z_2+1\right)$

$c\left(x,y,z\right)=\left(cx,cy,cz\right)$

(d) $\left(x_1,y_1,z_1\right)+\left(x_2,y_2,z_2\right)=\left(x_1+x_2+1,y_1+y_2+1,z_1+z_2+1\right)$

$c\left(x,y,z\right)=\left(cx+c-1,cy+c-1,cz+c-1\right)$

With each of these new definitions, is $R^3$ a vector space? Justify your answers.