Question

What inference can you draw if $\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{0}\mathrm{and}\overrightarrow{a}·\overrightarrow{b}=0.$

Answer 1

$$\begin{gathered} \text{Given:} \\ \left|\overrightarrow{a}\times \overrightarrow{b}\right|=\overrightarrow{0} \\ \Rightarrow \overrightarrow{a}=0 \\ \overrightarrow{b}\text{=0} \\ \therefore \overrightarrow{a}\text{∥}\overrightarrow{b} \\ \mathrm{Also}, \\ \overrightarrow{a}.\overrightarrow{b}=0 \\ \Rightarrow \left|\overrightarrow{a}\right|\left|\overrightarrow{b}\right|\cos \theta =0 \\ \Rightarrow \overrightarrow{a}=\overrightarrow{0}\text{or}\overrightarrow{b}\text{=}\overrightarrow{\text{0}}\text{or,}\overrightarrow{a}\text{⊥}\overrightarrow{b} \\ \mathrm{But}\overrightarrow{a}\mathrm{cannot}\mathrm{be}\mathrm{both}\mathrm{perpendicular}\mathrm{as}\mathrm{well}\mathrm{as}\mathrm{parallel}\mathrm{to}\overrightarrow{b}. \\ \therefore \left|\overrightarrow{a}\right|\text{=0} \\ \left|\overrightarrow{b}\right|\text{=0} \end{gathered}$$