Question

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

$v=3i-2j;w=4i+6j$

Short answer

The given vectors are orthogonal but not parallel.

Step-by-step solution

Step 1. Given Information 

The given vectors are$v=3i-2jandw=4i+6j.$

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 $0or\mathrm{\pi }.$

Let's find the angle

$$\begin{gathered} \cos \theta =\frac{\mathrm{v}·\mathrm{w}}{\left|\left|v\right|\right|\left|\left|w\right|\right|} \\ \cos \theta =\frac{\left(3i-2j\right)·\left(4i+6j\right)}{\left(\sqrt{3^2+{\left(-2\right)}^2}\right)\left(\sqrt{{\left(4\right)}^2+{\left(6\right)}^2}\right)} \\ \cos \theta =\frac{0}{26} \\ \cos \theta =0 \\ \theta ={\cos }^{-1}\left(0\right) \\ \theta =\frac{\mathrm{\pi }}{2} \end{gathered}$$

Thus, the given vectors are not parallel because the angle is neither$0nor\mathrm{\pi }.$

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

$$\begin{gathered} \mathrm{v}·\mathrm{w}=\left(3\mathrm{i}-2\mathrm{j}\right)·\left(4\mathrm{i}+6\mathrm{j}\right) \\ \mathrm{v}·\mathrm{w}=12-12 \\ \mathrm{v}·\mathrm{w}=0 \end{gathered}$$

Thus, the vectors are orthogonal.