Question

If $\overrightarrow{c}$is a unit vector perpendicular to the vectors $\overrightarrow{a}\mathrm{and}\overrightarrow{b},$write another unit vector perpendicular to $\overrightarrow{a}\mathrm{and}\overrightarrow{b}.$

Answer 1

$$\begin{gathered} \overrightarrow{c}\text{is a unit vector perpendicular to both}\overrightarrow{a}\text{and}\overrightarrow{b}\text{.} \\ \text{⇒}\overrightarrow{c}\text{=}\frac{\overrightarrow{a}\times \overrightarrow{b}}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|} \\ \text{⇒-}\overrightarrow{c}\text{=}\frac{\overrightarrow{b}\times \overrightarrow{a}}{\left|\overrightarrow{a}\times \overrightarrow{b}\right|} \\ \mathrm{Therefore},\text{-}\overrightarrow{c}\text{is perpendicular to}\overrightarrow{b}\text{and}\overrightarrow{a.} \\ \mathrm{Thus},-\overrightarrow{c}\text{is another unit vector perpendicular to}\overrightarrow{a}\text{and}\overrightarrow{b.} \end{gathered}$$