Khi nào thì \({\left( {\overrightarrow u .\overrightarrow v } \right)^2} = {\overrightarrow u ^2}.{\overrightarrow v ^2}?\)

Đáp án đúng là B

Ta có: \(\overrightarrow u .\overrightarrow v = \left| {\overrightarrow u } \right|.\left| {\overrightarrow v } \right|.c{\rm{os}}\left( {\overrightarrow u ,\overrightarrow v } \right)\)

\( \Leftrightarrow {\left( {\overrightarrow u .\overrightarrow v } \right)^2} = {\left[ {\left| {\overrightarrow u } \right|.\left| {\overrightarrow v } \right|.c{\rm{os}}\left( {\overrightarrow u ,\overrightarrow v } \right)} \right]^2} = {\overrightarrow u ^2}.{\overrightarrow v ^2}.c{\rm{o}}{{\rm{s}}^2}\left( {\overrightarrow u ,\overrightarrow v } \right)\)

Để \({\left( {\overrightarrow u .\overrightarrow v } \right)^2} = {\overrightarrow u ^2}.{\overrightarrow v ^2}\) thì \(c{\rm{o}}{{\rm{s}}^2}\left( {\overrightarrow u ,\overrightarrow v } \right) = 1 \Leftrightarrow \left[ \begin{array}{l}c{\rm{os}}\left( {\overrightarrow u ,\overrightarrow v } \right) = 1\\c{\rm{os}}\left( {\overrightarrow u ,\overrightarrow v } \right) = - 1\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}\left( {\overrightarrow u ,\overrightarrow v } \right) = {0^0}\\\left( {\overrightarrow u ,\overrightarrow v } \right) = {180^0}\end{array} \right.\)

Vậy khi góc giữa hai vecto \(\overrightarrow u ,\overrightarrow v \) là 0⁰ hoặc 180⁰ thì \({\left( {\overrightarrow u .\overrightarrow v } \right)^2} = {\overrightarrow u ^2}.{\overrightarrow v ^2}.\)