Find two vectors orthogonal to the three (in 4-space)?

Question

I have to find two vectors of norm 1 that are orthogonal to vectors:
(2,1,-4,0),
(-1,-1,2,2) , and
(3,2,5,4).
I am supossed to let the vectors be (x_1, x_2, x_3, x_4), and then solve a linear system. I should have a parameter in my solution, and determine that parameter using the norm 1 condition.
Could someone explain these steps in more detail please? I still don't understand what to do.

Answer 1

Let v1 = (2,1,-4,0), let v2 = the second given vector, and
let v3 = the third given vector.

Let X = (x_1, x_2, x_3, x_4).

You need the dot product of X with each of the three given vectors to be 0.

v1 dot X = 0
v2 dot X = 0
v3 dot X = 0

If you write out those dot products, you'll see that each is a linear expression in the four variables. Thus you have a homogeneous system of 3 linear equations in 4 variables. Row reduction will give you a solution containing a parameter.

Once you have a solution vector (in which each coordinate is a multiple of some parameter c), take the sum of the squares of the coordinates and set that equal to 1 ( so the solution has norm 1). Simplify, and you'll find yourself with a very simple quadratic equation to solve for c. There will be two solutions.

Answer 2

discover the vector projection of "u" onto "v" a = (u•v)/(v•v)v = (sixteen/7) = this is the 1st of the two vectors you seek for. to come across the different, subtract this from "u" b = u - a = (one million/7)<5, -4, one million> = <5/7, -4/7, one million/7>