Determine an orthogonal vector in $\mathbb{R}^4$

Question

Let $u_1$, $u_2$ be two vectors in $\mathbb{R}^4$:

$u_1=(2,0,1,1), u_2=(1,1,0,4)$

Determine a vector (different from null vector) that is orthogonal to both $u_1$ and $u_2$.

I'm stuck. Obviously I can't solve this by a system of equations as I would have 2 equations with 4 unknowns. How do I proceed?

Answer 1

Let $(x,y,z,w)$ be the vector orthogonal to both $u_1$ and $u_2$.

Then $2x+z+w=0$ and $x+y+4w=0$.

Now put $x=1$ in both equations. We will get,

$2+z=-w$ and $1+y=-4w$.

Take $w=1$. We will get,

$z=-3$ and $y=-5$.

So we have found a vector $(1,-5,-3,1)$ which is orthogonal to both $u_1$ and $u_2$.

Answer 2

Your system is underdetermined; there is a whole plane of orthgonal vectors to $u_1$ and $u_2$. Solving the system of equations will give you the equation for this plane.