If I have 3 vectors, a, b and c in 3D, I can check if they fulfill c = α a + β b (i.e. if they lie in a 2D plane) for some real parameter α and β by checking if ( a × b ) ⋅ c = 0. If I have 3 vectors in n dimensions, is there a similar, general formula to check if c = α a + β b?

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If I have 3 vectors, a, b and c in 3D, I can check if they fulfill   c  =  α  a  +  β  b (i.e. if they lie in a 2D plane) for some real parameter   α and   β by checking if   (  a  ×  b  )  ⋅  c  =  0. If I have 3 vectors in n dimensions, is there a similar, general formula to check if   c  =  α  a  +  β  b?

Lintynx · Accepted Answer

So I found something but I am not sure if it works. If we build the matrix   D  =  (  a  ,  b  ), which has dimensions   n  ×  2, then we can calculate   [  D  ⋅  (      D    T    D      )          −      1            D    T    ]  c. If this is equal to c, it means that the 3 vectors are coplanar. Does this hold in general?

duandaTed05 · Accepted Answer

c  =  α  a  +  β  b  if and only if  c  −                    (        a        ⋅        c        )        (        b        ⋅        b        )        −        (        a        ⋅        b        )        (        b        ⋅        c        )                    (        a        ⋅        a        )        (        b        ⋅        b        )        −        (        a        ⋅        b        )        (        a        ⋅        b        )              a  −                    (        a        ⋅        a        )        (        b        ⋅        c        )        −        (        a        ⋅        b        )        (        a        ⋅        c        )                    (        a        ⋅        a        )        (        b        ⋅        b        )        −        (        a        ⋅        b        )        (        a        ⋅        b        )              b  =  0Note that if the displayed equation holds, then   c  =  α  a  +  β  b with  α  =                    (        a        ⋅        c        )        (        b        ⋅        b        )        −        (        a        ⋅        b        )        (        b        ⋅        c        )                    (        a        ⋅        a        )        (        b        ⋅        b        )        −        (        a        ⋅        b        )        (        a        ⋅        b        )              ,    β  =                    (        a        ⋅        a        )        (        b        ⋅        c        )        −        (        a        ⋅        b        )        (        a        ⋅        c        )                    (        a        ⋅        a        )        (        b        ⋅        b        )        −        (        a        ⋅        b        )        (        a        ⋅        b        )            Going in the other direction, if   c  =  α  a  +  β  b, then  a  ⋅  c  =  (  a  ⋅  a  )  α  +  (  a  ⋅  b  )  β  ,    b  ⋅  c  =  (  a  ⋅  b  )  α  +  (  b  ⋅  b  )  βis a system of two linear equations in the two unknowns   α  ,  β and, if I haven&#039;t made any typos or silly errors in algebra, the solution to that system is given by the second display

If I have 3 vectors, a, b and c in 3D, I can check if they fulfill c=alpha a+beta b (i.e. if they lie in a 2D plane) for some real parameter alpha and beta by checking if (a xx b)*c=0. If I have 3 vectors in n dimensions, is there a similar, general formula to check if c=alpha a+beta b?

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