Trong không gian \(Oxyz\) cho hai điểm \(A\left( {0; - 2;3} \right),B\left( {1;0; - 1} \right)\). Tính sin góc hợp bởi hai véc tơ \(\overrightarrow {OA} ,\overrightarrow {OB} \).

Ta có:

\(\begin{array}{l}\overrightarrow {OA}  = \left( {0; - 2;3} \right) \Rightarrow \left| {\overrightarrow {OA} } \right| = \sqrt {{0^2} + {{\left( { - 2} \right)}^2} + {3^2}}  = \sqrt {13} \\\overrightarrow {OB}  = \left( {1;0; - 1} \right) \Rightarrow \left| {\overrightarrow {OB} } \right| = \sqrt {{1^2} + {0^2} + {{\left( { - 1} \right)}^2}}  = \sqrt 2 \end{array}\)

Suy ra $\left[ {\overrightarrow {OA} ,\overrightarrow {OB} } \right] = \left( {\left| {\begin{array}{*{20}{c}}\begin{array}{l} - 2\\0\end{array}&\begin{array}{l}3\\ - 1\end{array}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}\begin{array}{l}3\\ - 1\end{array}&\begin{array}{l}0\\1\end{array}\end{array}} \right|;\left| {\begin{array}{*{20}{c}}\begin{array}{l}0\\1\end{array}&\begin{array}{l} - 2\\0\end{array}\end{array}} \right|} \right) = \left( {2;3;2} \right)  $

$\Rightarrow \left| {\left[ {\overrightarrow {OA} ,\overrightarrow {OB} } \right]} \right| = \sqrt {{2^2} + {3^2} + {2^2}}  = \sqrt {17}$

Do đó \(\sin \left( {\overrightarrow {OA} ,\overrightarrow {OB} } \right) = \dfrac{{\left| {\left[ {\overrightarrow {OA} ,\overrightarrow {OB} } \right]} \right|}}{{\left| {\overrightarrow {OA} } \right|.\left| {\overrightarrow {OB} } \right|}} = \dfrac{{\sqrt {17} }}{{\sqrt {13} .\sqrt 2 }} = \sqrt {\dfrac{{17}}{{26}}} \)