Answer
Verification
Work Step by Step
To prove that commutative law of addition for vectors in $\mathbf{R}^4$
Let $x=(x_1,x_2,x_3,x_4) \in \mathbf{R}^4$
$\;\;\;\;, y=(y_1,y_2,y_3,y_4)\in \mathbf{R}^4$
We need to show that $x+y=y+x$
Consider $x+y=(x_1,x_2,x_3,x_4)+(y_1,y_2,y_3,y_4)$
$x+y=(x_1+y_1,x_2+y_2,x_3+y_3,x_4+y_4)$
Because all $x_i \; , \; y_i $ are real numbers
$x+y=(y_1+x_1,y_2+x_2,y_3+x_3,y_4+x_4)$
$x+y=(y_1,y_2,y_3,y_4)+(x_1,x_2,x_3,x_4)$
$\Rightarrow x+y=y+x$
Hence vectors in $\mathbf{R}^4$ satisfies commutative law of addition.
