Maths Vectors Question

Question

Show that, for all vectors, u and v, |u x v|^2 = |u|²|v|² - |u.v|²

Answer 1

|u × v|²

 

= (|u||v|sin θ)²

 

= (|u||v|)² sin² θ

 

= (|u||v|)² (1 - cos² θ)

 

= (|u||v|)² [1 - (|u · v|/(|u||v|))²]

 

= (|u||v|)² [1 - |u · v|²/(|u||v|)²]

 

= (|u||v|)² - |u · v|²

 

= |u|²|v|² - |u · v|²