Find Orthogonal complement
Olivia Zamora 
Let
$$U = Sp\{(3, 3, 1)\}$$
How can I find the Orthogonal complement ? I'm not sure how to calculate it.
In the book I'm learning from it's saying that I need to write the vectors of $U$ in $Ax = 0$ where the lines of $A$ are the vectors of $U$.
But since $U$ has only one vector I'm not sure how could this help me to find the orthogonal complement
$\endgroup$ 63 Answers
$\begingroup$Since $U$ has only one dimension, it is indeed true that $A$ will have only one line. Hence, the orthogonal complement $U^\perp$ is the set of vectors $\mathbf x = (x_1,x_2,x_3)$ such that \begin{equation} 3x_1 + 3x_2 + x_3 = 0 \end{equation} Setting respectively $x_3 = 0$ and $x_1 = 0$, you can find 2 independent vectors in $U^\perp$, for example $(1,-1,0)$ and $(0,-1,3)$. These generate $U^\perp$ since it is two dimensional (being the orthogonal complement of a one dimensional subspace in three dimensions). Hence, we can conclude that \begin{equation} U^\perp = \operatorname{Span}\{(1,-1,0),(0,-1,3)\}. \end{equation} Note that there would be many (infinitely many) other ways to describe $U^\perp$.
$\endgroup$ $\begingroup$The plane of equation $P \equiv 3x+3y+1z=0$.
It has dimension equal to $2$ and every vector $u=(x,y,z) \in P$ is such that $\langle u, v\rangle=0$ where $v=(3,3,1)$.
$\endgroup$ $\begingroup$As a complementary answer to the previous ones, we have that the orthogonal complement of $U$ is the set $V=U^\perp = \left\{ \vec{\mathbf v} = \begin{pmatrix} x\\ y\\z \end{pmatrix}: x,y,z \in \mathbb R\right\},$ such that: $$\vec{\mathbf u}\cdot \vec{\mathbf v} = 0, \quad{\text{ for every $\mathbf{\vec u}\in U$ }} .$$
Any vector $ \vec {\mathbf{u}} \in U$ will be of the form $a\cdot (3,3,1)=(3a,3a,a)$, where $a$ is a scalar in $\mathbb R$.
Having said that, we have: $$\begin{array}[t]{l} (3a, 3a, a) \cdot \begin{pmatrix} x \\ y \\z \end{pmatrix}= 0\\ 3ax + 3ay + az = 0\\ 3x + 3y + z = 0, \quad {\text{ since we want the above equation to hold for every $a\neq 0$.}} \end{array}$$
That means $V = U^\perp =\left\{ \begin{pmatrix} x\\ y \\z \end{pmatrix}\in \mathbb R^3: 3x+3y+z = 0\right\}$.
Any extra information is included in other answers.
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