The value of the expression $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$ is……….
The value of the expression $|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$ is……….
Official Solution
$|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$
$= |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2}{\sin ^2}\theta + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$
$= |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2}\left( {1 - {{\cos }^2}\theta } \right) + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$
$= |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2} - |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2}{\cos ^2}\theta + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$
$= |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2} - {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2} + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2}$
$|\overrightarrow {\rm{a}} \times \overrightarrow {\rm{b}} {|^2} + {(\overrightarrow {\rm{a}} \cdot \overrightarrow {\rm{b}} )^2} = |\overrightarrow {\rm{a}} {|^2}|\overrightarrow {\rm{b}} {|^2}$
No comments yet — start the discussion.