Bài 1. Chứng minh các công thức sau
a) \(\overrightarrow a .\,\overrightarrow b = {1 \over 2}\left( {|\overrightarrow a {|^2} + |\overrightarrow b {|^2} - \overrightarrow {|a} - \overrightarrow b {|^2}} \right)\);
b) \(\overrightarrow a .\,\overrightarrow b = {1 \over 4}\left( {|\overrightarrow a + \overrightarrow b {|^2} - |\overrightarrow a - \overrightarrow b {|^2}} \right)\).
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a) Ta có \(|\overrightarrow a - \overrightarrow b {|^2} = {(\overrightarrow a - \overrightarrow b )^2} = |\overrightarrow a {|^2} - 2\overrightarrow a \overrightarrow b + |\overrightarrow b {|^2}\)
\( \Rightarrow \,\,\,\overrightarrow a .\,\overrightarrow b = {1 \over 2}(|\overrightarrow a {|^2} + |\overrightarrow b {|^2} - |\overrightarrow a - \overrightarrow b {|^2})\)
b) Ta có \(|\overrightarrow a + \overrightarrow b {|^2} - |\overrightarrow a - \overrightarrow b {|^2} = {(\overrightarrow a + \overrightarrow b )^2} - {(\overrightarrow a - \overrightarrow b )^2}\)
\(\eqalign{
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = (\overrightarrow a + \overrightarrow b - \overrightarrow a + \overrightarrow b )(\overrightarrow a + \overrightarrow b + \overrightarrow a - \overrightarrow b ) = 4.\,\overrightarrow a .\,\overrightarrow b \cr
& \Rightarrow \,\,\overrightarrow a .\,\overrightarrow b = {1 \over 4}(|\overrightarrow a + \overrightarrow b {|^2} - |\overrightarrow a - \overrightarrow b {|^2}). \cr} \)