Chapter 9, Problem 66RE

To determine

To calculate: To use the vectors $v=-2i+j$ and $w=4i-3j$ to find,

  $\Vert 2v\Vert +3\Vert w\Vert$

Answer to Problem 66RE

The value of the vectorsis $2\sqrt{5}-15$ .

Explanation of Solution

Given information:

The vectors:

  $v=-2i+j$ and $w=4i-3j$

Formula used:

For the vectors by the additivity of $a+b$ in these components of ‘i’ added to ‘i’ and components of ‘j’ added to ‘j’,

Such as, $\left(a_{1}i+b_{1}j\right)+\left(a_{2}i+b_{2}j\right)=\left(a_{1}i+a_{2}i\right)+\left(b_{1}j+b_{2}j\right)$ .

The magnitude of $\Vert v\Vert +\Vert w\Vert$ is find by the help of formula:

  $\Vert v\Vert +\Vert w\Vert =\Vert \sqrt{a^2+b^2}\Vert +\Vert \sqrt{a^2+b^2}\Vert$

In the given equation or the given type of situation first multiply the vector $v$ by $2$ and then find the magnitude of $w$ vector separately $\Vert w\Vert$ then multiply by $3$ and then add both results.

Calculation:

Consider the vectors:

  $v=-2i+j$ and $w=4i-3j$

Put the value of the vectors in the given expression $\Vert 2v\Vert -3\Vert w\Vert$ .

Now $v=-2i+j$ and $w=4i-3j$ .

Now write the vectors $v=-2i+j$ and $w=4i-3j$ in $\Vert 2v\Vert -3\Vert w\Vert$ .

To write the vectors $v=-2i+j$ and $w=4i-3j$ in this form first we find $2v$ that means first multiply the vector $v$ by $2$ and then find the magnitude of the vector $v$ .

Then to find the vector $w,w=4i-3j$ first find the magnitude of $w$ vector then multiply by $3$ .

Now firstly find the $\Vert 2v\Vert$ ,

  $\begin{array}{l}\Vert 2v\Vert =\Vert 2\left(-2i+j\right)\Vert \\ =\Vert -4i+2j\Vert \\ =\sqrt{{\left(-4\right)}^2+{\left(2\right)}^2}\\ =\sqrt{16+4}\end{array}$

Thus the value is:

  $\begin{array}{c}\sqrt{{\left(-4\right)}^2+{\left(2\right)}^2}=\sqrt{16+4}\\ =\sqrt{20}\\ =2\sqrt{5}\end{array}$

Now find $3\Vert w\Vert$ ,

First find magnitude of $w$ vector,

  $\begin{array}{c}\Vert w\Vert =\sqrt{{\left(4\right)}^2+{\left(-3\right)}^2}\\ =\sqrt{16+9}\\ =\sqrt{25}\\ =5\end{array}$

Now multiply the $\Vert w\Vert$ by $3$ , that means $3\Vert w\Vert$ .

  $3\cdot 5=15$

Now put these values in the given expression $\Vert 2v\Vert -3\Vert w\Vert$ .

  $\Vert 2v\Vert -3\Vert w\Vert =2\sqrt{5}-15$ .

Thus, the value of the magnitude of vectors are $2\sqrt{5}-15$ .