# Thread: Find Unit vector parallel to xy plane and perp. to a vector

1. ## Find Unit vector parallel to xy plane and perp. to a vector

Find all unit vectors parallel to the xy-plane and perpendicular to the vector [1, -2, 2].

so
$
a=[1,-2,2]$

$v=[x,y,0]$
Using Dot Product
$adotb=0$
$1x-2y+2(0)=0$

After that i rearrange and get stuck with x-x=0... I don't see any other ways to do this.

2. Originally Posted by JimmyRP
Find all unit vectors parallel to the xy-plane and perpendicular to the vector [1, -2, 2].
I find two. The key is the fact they must be units.
$\pm \left\langle {\frac{2}{{\sqrt 5 }},\frac{1}{{\sqrt 5 }},0} \right\rangle$

3. so you got

$

1x-2y+2(0)=0
$

$

x-2y=0
$

so

$

x=2y
$

so the vector
$

\left\langle {2y,y,0} \right\rangle
$

is perpendicular to it but we need it to be a unit vector

its magnitude is $\sqrt{4y^2+y^2}=\sqrt{5}|y|$

so our unit vector is $\frac{v}{\sqrt{5}|y|}$
=
$

\left\langle {\frac{2y}{{\sqrt 5 }|y|},\frac{y}{{\sqrt 5 }|y|},0} \right\rangle
$

we get +ve and -ve depending on whether y>0 or <0 i.e two opposite directions which makes sense

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# the vector pararlael to x-y palne

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