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

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?

• Trial and error is easiest for this one. Try one that is just orthogonal to the first vector and then modify the second component of it till it works for the second vector as well. Nov 23, 2016 at 4:56
• @ravi Solving this problem systematically instead of guessing is rather easy, too.
– amd
Nov 23, 2016 at 8:17

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 $u_2$.

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.