Answer
By writing both sides of the equation in component form and using the following basic definitions:
1. definition of vector addition
2. definition of scalar multiplication
3. distributive property of real numbers
one can verify the properties in the question by showing that both sides of the equation are equal.
Work Step by Step
General strategy for both parts: compute both the left hand side (LHS) and the right hand (RHS) side of the equation in component form and show that they are equal.
(a) LHS:
First, from the definition of vector addition:
$u + v$ = ($u_{1}$ + $v_1$, $u_2$ + $v_2$, ... , $u_n$ + $v_n$)
Therefore, from the definition of vector addition again:
$(u + v) + w$
= ($u_{1}$ + $v_1$, $u_2$ + $v_2$, ... , $u_n$ + $v_n$) + ($w_1$, $w_2$, ..., $w_n$)
= ($u_{1}$ + $v_1$ + $w_1$, $u_2$ + $v_2$ + $w_2$, ... , $u_n$ + $v_n$ + $w_n$) .......... (1)
RHS:
First, from the definition of vector addition:
$v + w$ = ($v_{1}$ + $w_1$, $v_2$ + $w_2$, ... , $v_n$ + $w_n$)
Therefore, from the definition of vector addition again:
$u + (v + w)$
= ($u_{1}, u_2, u_3$) + ($v_1 +w_1, v_2 + w_2, ... , v_n + w_n$)
= ($u_{1}$ + $v_1$ + $w_1$, $u_2$ + $v_2$ + $w_2$, ... , $u_n$ + $v_n$ + $w_n$) .......... (2)
Since equation (1) is the same as equation (2) the property is verified.
(b) LHS:
First, from definition of vector addition:
$u + v$ = ($u_{1} + v_1, u_2 + v_2, ... , u_n + v_n$)
Second, from definition of scalar multiplication:
$c(u + v) = (c(u_{1} + v_1), c(u_2 + v_2), ... , c(u_n + v_n))$
Third, from the distributive property of real numbers i.e. if $x, y, z$ are all real numbers then $z(x + y) = zx + zy$ we find:
$c(u + v)
= (cu_{1} + cv_1, cu_2 + cv_2, ... , cu_n + cv_n)$ .......... (3)
RHS:
First, from definition of scalar multiplication:
$c\times u = (cu_1, cu_2, ... cu_n)$
$c\times v = (cv_1, cv_2, ... cv_n)$
Therefore, from the definition of vector addition:
$c\times u + c\times v $
$= (cu_1, cu_2, ... cu_n) + (cv_1, cv_2, ..., cv_n)$
$= (cu_1 + cv_1, cu_2 + cv_2, ... , cu_n + cv_n)$ .......... (4)
Since equation (3) is the same as equation (4) the property is verified.
