Find a nonzero vector xrightarrow perpendicular to the vecto

Find a nonzero vector x^rightarrow perpendicular to the vectors v^rightarrow = [-9 -6 -2 -7] and u^rightarrow = [9 7 8 -5]. x^rightarrow = []

Solution

Let x = (p,q,r,s)^T be a vector perpendicular to v and u. Then x.v = 0 or, -9p-6q-2r-7s = 0..(1)

Also, x.u = 0 or, 9p+7q+8r-5s = 0….(2)

On adding the above 2 equations, we get -9p-6q-2r-7s +9p+7q+8r-5s = 0 or, q+6r-12s = 0 so that q = -6r +12s. Then 9p = -6q-2r-7s = -6(-6r+12s)-2r-7s = 36r-72s -2r-7s = 34r -79s so that p =34r/9- 79s/9 .Then x = (p,q,r,s)^T = (34r/9-79s/9, -6r +12s, r, s)^T.If, r=1 and s = 1, then x = (-5, 6, 1,1)^T