Question

Prove part (a) of Theorem 10.8 for vectors in $R^3$; that is, show that for $u=\left(u_1,u_2,u_3\right)$ and $v=\left(v_1,v_2,v_3\right)$, $u+v=v+u$

Short answer

It is proven that$u+v=v+u$

Step-by-step solution

Given

Vectors$u\mathrm{and}v$in$R^3$.

Proof

Let us take two vectors,

$$\begin{gathered} u=1i+2j+3k\mathrm{and}v=4i+5j+6k \\ LHS=u+v \\ =\left(1i+2j+3k\right)+\left(4i+5j+6k\right) \\ =\left(1i+4i\right)+(2j+5j)+(3k+6k) \\ =5i+7j+9k \\ RHS=v+u \\ =\left(4i+5j+6k\right)+\left(1i+2j+3k\right) \\ =\left(1i+4i\right)+(2j+5j)+(3k+6k) \\ =5i+7j+9k \end{gathered}$$

It is observed that, $LHS=RHS$

Hence, proved.