Question

In Problems 19–22,

Which two vectors are orthogonal?

Short answer

$v_2$and $v_3$ are orthogonal.

Step-by-step solution

Step 1. Given information

The given vectors are

Step 2. Find the vectors which are orthogonal.

For finding vectors which are parallel, the angle between vectors should be $90°$

We will find the angles between vectors.

For finding the angles between any two vectors $\overrightarrow{a}$and $\overrightarrow{b}$,

$\cos \theta =\frac{\overrightarrow{a}·\overrightarrow{b}}{||\overrightarrow{a}||||\overrightarrow{b}||}$

Step 3. Find the angle between v1 and v2.

$$\begin{gathered} \cos \theta =\frac{v_1·v_2}{||v_1||||v_2||} \\ =\frac{-12+\left(-36\right)}{\left(\sqrt{4^2+6^2}\right)\left(\sqrt{{\left(-3\right)}^2+{\left(-6\right)}^2}\right)} \\ =\frac{-48}{\sqrt{52}\times \sqrt{45}} \end{gathered}$$

These are not orthogonal.

Step 4. Find the angle between v2 and v3.

$$\begin{gathered} \cos \theta =\frac{v_2·v_3}{||v_2||||v_3||} \\ =\frac{24+\left(-24\right)}{\left(\sqrt{{\left(-8\right)}^2+4^2}\right)\left(\sqrt{{\left(-3\right)}^2+{\left(-6\right)}^2}\right)} \\ =0 \\ \theta =90° \end{gathered}$$

These are orthogonal.

Step 5. Find the angle between v3 and v4.

$$\begin{gathered} \cos \theta =\frac{v_3·v_4}{||v_3||||v_4||} \\ =\frac{-80+60}{\left(\sqrt{{\left(-8\right)}^2+4^2}\right)\left(\sqrt{{10}^2+{15}^2}\right)} \\ =\frac{-20}{\sqrt{80}\times \sqrt{325}} \end{gathered}$$

These are not orthogonal.

Step 6. Find the angle between v1 and v4.

$$\begin{gathered} \cos \theta =\frac{v_1·v_4}{||v_1||||v_4||} \\ =\frac{40+90}{\left(\sqrt{4^2+6^2}\right)\left(\sqrt{{10}^2+{15}^2}\right)} \\ =\frac{130}{\sqrt{52}\times \sqrt{325}} \\ =1 \\ \theta =0° \end{gathered}$$

These are not orthogonal.