Question

If three points with position vectors $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}$ are collinear, then $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.$

Answer 1

If $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}$ are collinear,

Let points A, B, C be collinear, where position vectors are $\overrightarrow{a},\overrightarrow{b}\mathrm{and}\overrightarrow{c}$ respectively, since $\overrightarrow{AB}\mathrm{and}\overrightarrow{BC}$ are parallel vectors,

$$\begin{gathered} \mathrm{i}.\mathrm{e}\overrightarrow{AB}\times \overrightarrow{BC}=0 \\ \mathrm{i}.\mathrm{e}\left(\overrightarrow{b}-\overrightarrow{a}\right)\times \left(\overrightarrow{c}-\overrightarrow{b}\right)=0 \\ \mathrm{i}.\mathrm{e}\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{b}\times \left(-\overrightarrow{b}\right)+\left(-\overrightarrow{a}\right)\times \overrightarrow{c}+\overrightarrow{a}\times \overrightarrow{b}=0 \\ \mathrm{i}.\mathrm{e}\overrightarrow{b}\times \overrightarrow{c}+0+\overrightarrow{c}\times \overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}=0 \\ \mathrm{i}.\mathrm{e}\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=0 \end{gathered}$$