Chapter 6, Problem 56RE

To determine

Find the component form of $w$ and sketch the specified vector operations geometrically, where $u=6d-5J$ and $v=101+3j$

Answer to Problem 56RE

The vector u as a linear combination of the standard unit vector is be $-1i+5j$

Explanation of Solution

Given:

  $w=4u-5v$

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's consider the following

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

Find the component form of $w=\frac{1}{2}v$

The value of $w=\frac{1}{2}v$ is

  $\begin{array}{cc}\frac{1}{2}v& =\frac{1}{2}\times (10i+3j)\\ & =\left(\frac{1}{2}\times 10i+\frac{1}{2}\times 3j\right)\\ & =5i+1.5j\end{array}$

Thus the component form of $w=5i+1.5j$

And the graph of

  $w=\frac{1}{2}v$ given below