Let vec(a) ,vec(b) ,vec(c) be non-complanar vectors so that vec(a) xx vec(b) , vec(b) , vec(c) xx vec(a) on the same plane. How can I show that vec(a) is perpendicular to vec(b) ?

Paula Cameron

Paula Cameron

Answered question

2022-11-11

Let a , b , c be non-complanar vectors so that a × b , b , c × a on the same plane. How can I show that a is perpendicular to b ?

Answer & Explanation

Maffei2el

Maffei2el

Beginner2022-11-12Added 20 answers

We can (probably) write
b = λ ( a × b ) + μ ( c × a )
for some scalars λ , μ as the three vectors are co-planar. Now the inner product with a of this expression will be 0, as the inner product is linear and a ( a × b ) = 0 and a ( c × a ) = 0
The previous will only fail if c × a and ( a × b ) are dependent;

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