Chapter 4.11, Problem 1AP

Additional Problems

For problem 1-2, let $r$ and $s$ denote scalars and let $v$ and $w$ denote vectors in ${ℝ}^5$.

Prove that $\left(r+s\right)v=rv+sv$.

To determine

To prove:

That $\left(r+s\right)v=rv+sv$.

Answer to Problem 1AP

Solution:

It is verified that $\left(r+s\right)v=rv+sv$.

Explanation of Solution

Given:

Let $r,s$ be scalars and $v$ is a vector quantity.

Calculation:

Suppose $v=\left(v_1,v_2\right)\in {ℝ}^5$.

Then,

Substitute $\left(v_1,v_2\right)$ for $v$ in above expression.

$\begin{array}{rll}\left(r+s\right)v& =& \left(r+s\right)\left(v_1,v_2\right)\\ & =& \left(\left(r+s\right)v_1,\left(r+s\right)v_2\right)\\ & =& \left(r{v}_1+s{v}_1,r{v}_2+s{v}_2\right)\\ & =& r\left(v_1,v_2\right)+s\left(v_1,v_2\right)\\ & =& rv+sv\end{array}$

Thus, it is verified that $\left(r+s\right)v=rv+sv$.

Conclusion:

Hence, it is verified that $\left(r+s\right)v=rv+sv$.