Find a nonzero vector x perpendicular to the vectors v 6 6

Find a nonzero vector x perpendicular to the vectors v = [-6 -6 -7 3] and u = [4 8 -3 -4]. Set up a system of linear equations that the components of x satisfy.

Solution

Let, x=(a,b,c,d)

So

x.v=0 which gives:

-6a-6b-7c+3d=0

x.u=0 gives:

4a+8b-3c-4d=0

Two equations and 4 variables so we should be able to eliminate 2 variables and get 2 free variables.

2* first equation+3* second equatoin gives:

12b-23c-6d=0

Hence,

d=2b-23c/6

4a+8b-3c-4d=0

Substituting here expresion for d gives:

4a+8b-3c-4(2b-23c/6)=0

4a+8b-3c-8b+46c/3=0

4a+37c/3=0

a=-37c/12

SO here, c and b are the free variables.

Set, c=0 and b=1

We get:

x=(0,1,0,2)^T

This is one vector perpendicular to both equations.