# Thread: Find all vectors N that are orthogonal to a b and c

1. ## 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?

2. The normal vector to all of these vectors will have the same equation as the plane that contains all three vectors.

3. Originally Posted by technoboy
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