Question

Let $c$and $d$be a scalars and let $v$ be a vector in $R^3$. Show that the following distributive property holds: role="math" localid="1663644928733" $(c+d)v=cv+dv$

Short answer

It is proved that,$(c+d)v=cv+dv$

Step-by-step solution

Consider a vector

Let us assume a vector in $R^3$

$v=i+j+k$

The scalers$c=1\mathrm{and}d=2$

Proof

Let us consider the LHS of the given expression,

$$\begin{gathered} \mathrm{LHS}=(c+d)v \\ =(1+2)(i+j+k) \\ =3(i+j+k) \\ =3i+3j+3k \\ \mathrm{RHS}=cv+dv \\ =1(i+j+k)+2(i+j+k) \\ =i+j+k+2i+2j+2k \\ =3i+3j+3k \end{gathered}$$

Since, $\mathrm{LHS}=\mathrm{RHS}$

Hence, proved.