Answer
Proof
Work Step by Step
To prove vectors in $\mathbf{R}^4$ satisfies associativity under addition
Let $x=(x_1,x_2,x_3,x_4) \in \mathbf{R}^4$
$\;\;\;\;,y=(y_1,y_2,y_3,y_4) \in \mathbf{R}^4$
$\;\;\;\;,z=(z_1,z_2,z_3,z_4) \in \mathbf{R}^4$
Enough to show that $x+(y+z)=(x+y)+z$
Consider
$x+(y+z)=(x_1,x_2,x_3,x_4)+[(y_1,y_2,y_3,y_4)+(z_1,z_2,z_3,z_4)]$
$x+(y+z)=(x_1,x_2,x_3,x_4)+(y_1+z_1,y_2+z_2,y_3+z_3,y_4+z_4)$
$x+(y+z)=(x_1+y_1+z_1,x_2+y_2+z_2,x_3+y_3+z_3,x_4+y_4+z_4)$
Because all $x_i , y_i , z_i$ are real numbers
$\Rightarrow x+(y+z)=(x_1+y_1,x_2+y_2,x_3+y_3,x_4+y_4)+(z_1,z_2,z_3,z_4)$
$\Rightarrow x+(y+z)=(x+y)+z$
Hence vectors in $\mathbf{R}^4$ satisfies associativity under addition.
