Let R4 have the Euclidean inner product Find two unit vector

Let R4 have the Euclidean inner product. Find two unit vectors that are orthogonal to the three vectors u= (5, 4, -5, 0), v = (-1, -1, 1, 1), and w = (4, 3, 3, 5). Give the exact answer using fractions and the sqrt() function. x_1 = (-32/(2314)^0.5, x_2 =

Solution

u * x = 0 ==> 5a + 4c - 5c = 0
v * x = 0 ==> -a -b + c + d = 0
w * x = 0 ==> 4a + 3b + 3c + 5d = 0.

This gives a system of linear equations with matrix
[5 4 -5 0]
[-1 -1 1 1]
[4 3 3 5].

Row reducing yields

[1 0 0 -32/7]
[0 1 0 5]
[0 0 1 -4/7].

So, x = [-32 35 -4 7] (or any scalar multiple thereof).

Normalizing x yields the unit vectors
± [-32 35 -4 7] / sqrt(32^2 + 35^2 + 4^2 + 7^2)
= ± [-34, 44, -6, 11] / 48.10.

(The \"±\" denotes the direction of the unit vector; that\'s why there are two of them.)

I hope this helps!