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

• Oct 5th 2009, 01:26 PM
JimmyRP
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
$\displaystyle a=[1,-2,2]$
$\displaystyle v=[x,y,0]$
Using Dot Product
$\displaystyle adotb=0$
$\displaystyle 1x-2y+2(0)=0$

After that i rearrange and get stuck with x-x=0... I don't see any other ways to do this.
• Oct 5th 2009, 01:44 PM
Plato
Quote:

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.
$\displaystyle \pm \left\langle {\frac{2}{{\sqrt 5 }},\frac{1}{{\sqrt 5 }},0} \right\rangle$
• Oct 5th 2009, 03:25 PM
Krahl
so you got

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

$\displaystyle x-2y=0$

so

$\displaystyle x=2y$
so the vector
$\displaystyle \left\langle {2y,y,0} \right\rangle$
is perpendicular to it but we need it to be a unit vector

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

so our unit vector is $\displaystyle \frac{v}{\sqrt{5}|y|}$
=
$\displaystyle \left\langle {\frac{2y}{{\sqrt 5 }|y|},\frac{y}{{\sqrt 5 }|y|},0} \right\rangle$

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