Question

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

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

Short answer

The given vectors are parallel but not orthogonal.

Step-by-step solution

Step 1. Given Information

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

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 $0\mathrm{or}\mathrm{\pi }.$

So,

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

Thus,vand ware parallel.

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 $\mathrm{v}·\mathrm{w}=0.$

So,

$$\begin{gathered} \mathrm{v}·\mathrm{w}=\left(2\mathrm{i}+3\mathrm{j}\right)·\left(-4\mathrm{i}-6\mathrm{j}\right) \\ \mathrm{v}·\mathrm{w}=-8-18 \\ \mathrm{v}·\mathrm{w}=-26 \end{gathered}$$

Thus, the vectors are not orthogonal.