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- 1. Let u and v be non-zero vectors in 3-D space. Establish ||u×v||^2 = ||u||^2||v||^2 - (u·v)^2 Show work Solution (u.v) 2 = ||u|| 2 ||v|| 2 cos 2 ||u|| 2 ||v|| 2 -(u.v) 2 = ||u|| 2 ||v|| 2 -||u|| 2 ||v|| 2 cos 2 = ||u|| 2 ||v|| 2 (1-cos 2 ) = ||u|| 2 ||v|| 2 sin 2 =>||u|| 2 ||v|| 2 -(u.v) 2 = (||u||.||v||.sin) 2 = ||uxv|| 2 ||u|| 2 ||v|| 2 -(u.v) 2 = ||uxv|| 2