Q. 59 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Consider vectors

Let us consider vectors are,

$$\begin{gathered} u=2i+j+k \\ v=i+3j+5k \\ \mathrm{and} c=2 \end{gathered}$$

Step 2: Proof

Consider the LHS of the given expression,

$$\begin{gathered} \mathrm{LHS}=c(u+v) \\ =2(2i+j+k+i+3j+5k) \\ =2(3i+4j+6k) \\ =6i+8j+12k \end{gathered}$$

$$\begin{gathered} \mathrm{RHS}=cu+cv \\ =2(2i+j+k)+2(i+3j+5k) \\ =4i+2j+2k+2i+6j+10k \\ =6i+8j+12k \end{gathered}$$

It is observed that, $\mathrm{LHS}=\mathrm{RHS}$

Hence, proved.