Question

Find each quantity if $\mathit{v}=3\mathit{i}\mathit{-}\mathit{5}\mathit{j}\mathit{,}\mathit{}\mathit{w}\mathit{=}\mathit{-}\mathit{2}\mathit{i}\mathit{+}\mathit{3}\mathit{j}$.

$\left|\left|\mathit{v}\right|\right|-\left|\left|\mathit{w}\right|\right|$

Short answer

The quantity of $\left|\left|\mathit{v}\right|\right|-\left|\left|\mathit{w}\right|\right|$is $\sqrt{34}-\sqrt{13}$.

Step-by-step solution

Step 1. Given information.

Consider the given question,

$$\begin{gathered} \mathit{v}=3\mathit{i}\mathit{-}\mathit{5}\mathit{j} \\ \mathit{w}\mathit{=}\mathit{-}\mathit{2}\mathit{i}\mathit{+}\mathit{3}\mathit{j} \end{gathered}$$

The magnitude of the vector $\mathit{v}=a_1\mathit{i}+b_1\mathit{j}$,

$\left|\left|\mathit{v}\right|\right|=\sqrt{{\left(a_1\right)}^2+{\left(b_1\right)}^2}$

Compare localid="1646752350191" $\mathit{v}=3\mathit{i}\mathit{-}\mathit{5}\mathit{j}$with the standard form localid="1646752360024" $\mathit{v}=a_1\mathit{i}+b_1\mathit{j}$,

$$\begin{gathered} a_1=3, \\ {b}_1=-5 \end{gathered}$$

Then,

$$\begin{gathered} \left|\left|\mathit{v}\right|\right|=\sqrt{{\left(3\right)}^2+{\left(-5\right)}^2} \\ \left|\left|\mathit{v}\right|\right|=\sqrt{9+25} \\ \left|\left|\mathit{v}\right|\right|=\sqrt{34} \end{gathered}$$

Step 2. Compare w with the standard form.

Compare $\mathit{w}\mathit{=}\mathit{-}\mathit{2}\mathit{i}\mathit{+}\mathit{3}\mathit{j}$ with the standard form $\mathit{v}=a_1\mathit{i}+b_1\mathit{j}$,

$$\begin{gathered} a_1=-2 \\ {b}_1=3 \end{gathered}$$

Then,

$$\begin{gathered} \left|\left|\mathit{w}\right|\right|=\sqrt{{\left(-2\right)}^2+3^2} \\ \left|\left|\mathit{w}\right|\right|=\sqrt{4+9} \\ \left|\left|\mathit{w}\right|\right|=\sqrt{13} \end{gathered}$$

Therefore, we can write,

$\left|\left|\mathit{v}\right|\right|-\left|\left|\mathit{w}\right|\right|=\sqrt{34}-\sqrt{13}$