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Find two vectors orthogonal to $v = [1,2,0]$

Usually i see questions with asking you two find given two vectors find two orthogonal vectors for it. Then you would use cross product and then use the result to find the unit vector.

What i do not understand is how would i do this for a single vector if i'm trying to find two vectors orthogonal to it

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2 Answers

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Denote "$v$ is orthogonal to $w$" by $v\ \bot\ w$. Then we define orthogonality by $$v\ \bot\ w \iff v\cdot w=0$$ where $v\cdot w$ is the dot product of $v,w \in \Bbb R^n$.

So a vector $(x,y,z)$ is orthogonal to $v$ if $$(x,y,z) \cdot (1,2,0)=x+2y=0$$

Clearly there are no restrictions on $z$ so you can pick any value of $z$. But $x$ and $y$ are related by the equation $x=-2y$.

So just pick any values of $x$'s (or $y$'s) and $z$'s and use that equation to find the last coordinate of suitable vectors.

For instance $(2,-1,0)$, $(-10,5,3)$, $(2\pi,-\pi, e)$, $(0,0,\frac {11}{3})$, etc are all vectors orthogonal to $v$. Confirm this for yourself.

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"Orthogonal" also means perpendicular so all you would need is two vectors perpendicular to $v = [1,2,0]$, so the answers could be -v which is $v = [-1, -2, 0]$ and $[0,0,1]$

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