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Math Help - Find all vectors N that are orthogonal to a b and c

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    Question Find all vectors N that are orthogonal to a b and c

    Find all vectors N that are orthogonal to a b and c where
    a =[1 2 3 4]; b =[-1 2 -3 -4]; c =[2 4 6 10]

    my idea is, since the dot product between to vectors is 0, we could do the same thing for all the vectors such that

    n*a = 0
    n*b = 0
    n*c = 0

    and then implement them into a system of linear equations such that
    n_1 + 2n_2 + 3n_3 +4n_4 = 0
    -n_1 + 2n_2 - 3n_3 - 4n_4 = 0
    2n_1 + 4n_2 + 6n_3 + 10n_4 = 0

    where n_(1-4) are all the variables of the vector N such that

    N = [n_1 n_2 n_3 n_4]

    would this be a valid way to approach this?
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  2. #2
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    The normal vector to all of these vectors will have the same equation as the plane that contains all three vectors.
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  3. #3
    MHF Contributor FernandoRevilla's Avatar
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    Quote Originally Posted by technoboy View Post
    would this be a valid way to approach this?

    Right. Solving you'll obtain:


    N \equiv\; \begin{bmatrix}{n_1}\\{n_2}\\{n_3}\\{n_4}\end{bmat  rix}=\lambda \begin{bmatrix}{-3}\\{0}\\{1}\\{0}\end{bmatrix}\quad (\lambda \in \mathbb{R})


    Fernando Revilla
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