Question

In Problems 50–52, determine whether v and w are parallel, orthogonal, or neither.

$v=-2i+2j;w=-3i+2j$

Short answer

The given vectors are neither parallel nor orthogonal.

Step-by-step solution

Step 1. Given Information 

The given vectors are$v=-2i+2jandw=-3i+2j.$

Step 2. Determining whether v and w are parallel or not 

To determine whether the two vectors are parallel or not, we have to find the angle between them because two vectors v and w are said to be parallel if the angle between them is $0or\mathrm{\pi }.$

Ley's find the angle

$$\begin{gathered} \cos \theta =\frac{\mathrm{v}·\mathrm{w}}{\left|\left|v\right|\right|\left|\left|w\right|\right|} \\ \cos \theta =\frac{\left(-2i+2j\right)·\left(-3i+2j\right)}{\left(\sqrt{{\left(-2\right)}^2+2^2}\right)\left(\sqrt{{\left(-3\right)}^2+2^2}\right)} \\ \cos \theta =\frac{10}{\sqrt{104}} \\ \theta ={\cos }^{-1}\left(\frac{10}{\sqrt{104}}\right) \\ \theta \approx 11.3° \end{gathered}$$

Thus, the given vectors are not parallel because the angle is neitherrole="math" localid="1647001247424" $0nor\mathrm{\pi }.$

Step 3. Determining whether v and w are orthogonal or not 

Two vectors v and w are said to be orthogonal if and only if $\mathrm{v}·\mathrm{w}=0.$

Let's find the dot product

$$\begin{gathered} \mathrm{v}·\mathrm{w}=\left(-2i+2j\right)·\left(-3i+2j\right) \\ \mathrm{v}·\mathrm{w}=6+4 \\ \mathrm{v}·\mathrm{w}=10 \end{gathered}$$

Thus, the vectors are not orthogonal.