Q. 61 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Consider a vector

Let us assume a vector in $R^3$

$v=i+j+k$

The scalers $c=1 \mathrm{and} d=2$

Step 2: Proof

Let us consider the LHS of the given expression,

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

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

Hence, proved.