Let u and v be non-zero vectors in 3-D space- Establish --uv--^2 - --.docx

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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 .

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

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