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U is a subspace of $\mathbb{R}^3$ $$U=\{x\in\mathbb{R}^3|x_1+x_2+x_3=0\}$$

Compute the ortogonal complement of the subspace.

How can I do this? I have searched for hours. Thank You!!!

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    Find vectors which span the subspace. Then find all vectors orthogonal to them. – 2017-01-13
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    Can you write the defining equation $x_1 + x_2 + x_3 = 0$ as a matrix-vector equation? – 2017-01-13
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    Hint: $x_1+x_2+x_3=0$ is true if and only if $x_1\cdot t+x_2\cdot t+x_3\cdot t=0$ for all $t$, which gives you something that $U$ is the orthogonal complement of. Do you know that "orthogonal complement" is a symmetric relation between subspaces? – 2017-01-13
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    @HenningMakholm just so no one gets confused, for future readers: it happens to be a symmetric relation here because subspaces of $\Bbb R^n$ are closed. In general, for subspaces it only holds that $U^{\perp \perp} = \overline U$. – 2017-01-13

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The basis for $U$ is $\{a=(-1,1,0),b=(-1,0,1)\}$. Let $x=(x_1,x_2,x_3)$ be in ${U}^{\perp}$. Then $\langle x,a \rangle=-x_1+x_2=0$ and $\langle x, b \rangle=-x_1+x_3=0$. Thus $x_1=x_2=x_3$. Hence $U^{\perp}=\{(x_1,x_2,x_3): x_1=x_2=x_3\}$

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Hint: Write the defining equation for $U$ as $(1,1,1)^T\cdot(x_1,x_2,x_3)^T=0$. What does this say about the relationship of elements of $U$ to the vector $(1,1,1)^T$, and so also to its span? (This hint is trying to lead you to the realization that a basis for the orthogonal complement of $U$ can be read from its defining equation.)

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Note that $U$ has $2$ degrees of freedom, i.e., $\dim U$=2, so you can denote $x_2=t$ and $x_3 = s$, thus every vector in $U$ is given by $(-t-s, t, s)$ or equivalently can be expressed as a linear combination of $(-1,1,0)$ and $(-1,0,1)$. As such $U^\perp$ is all vectors that are orthogonal to the space (plane) that is spanned by these two vectors. Hence, such vector have to be orthogonal to each one of them. Denote such vector by $(a,b,c)$, thus it should satisfy $(a,b,c)\cdot(-1,1,0)=-a+b=0$ and $(a,b,c)\cdot(-1,0,1)=-a+c=0$. Denote $x_1 = t$, then such vectors are of the form $t(1,1,1)$, $t \in \mathbb{R}$ or $$ U^\perp = \{\mathrm{x}\in \mathbb{R}^3|x_1=x_2=x_3\}. $$