Chapter 6, Problem 54RE

To determine

Find the component form of w and sketch the specified vector components geometrically.

Answer to Problem 54RE

The component form of $w=[-26i-35j]$

Explanation of Solution

Given:

  $w=\frac{1}{2}v$

Let $u=\left(u_1,u_2\right)$ and $v=\left(v_1,v_2\right)$ be the vectors and let $k$ be a scalar

Then

  $\begin{array}{cc}u+v& =\left(u_1+v_1,u_2+v_2\right)\\ ku& =\left(k{u}_1,k{u}_2\right)\\ -v& =\left(-v_1,-v_2\right)\\ u-v& =\left(u_1-v_1,u_2-v_2\right)\end{array}$

Let us consider the following

  $\begin{array}{l}u=6i-5j,\text{and}\hfill \\ v=10i+3j\hfill \end{array}$

Then find the component form of $w=4u-5v$

The value of $4u$ is

  $\begin{array}{cc}4u& =4\times (6i-5j)\\ & =24i-20j\end{array}$

And the value of $5v$ is

  $\begin{array}{cc}5v& =5\times (10i+3j)\\ & =(5\times 10i+5\times 3j)\\ & =50i+15j\end{array}$

Then the value of $w=4u-5v$ is

  $\begin{array}{cc}w& =4u-5v\\ & =(24i-20j)-(50i+15j)\\ & =24i-20j-50i-15j\\ & =-26i-35j\end{array}$

Thus the component form of $w=[-26i-35j]$

And the graph of $w=4u-5v$ given below