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

**JimmyRP** · October 5th 2009, 01:26 PM

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.

**Plato** · October 5th 2009, 01:44 PM

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.
$\pm \left\langle {\frac{2}{{\sqrt 5 }},\frac{1}{{\sqrt 5 }},0} \right\rangle$

**Krahl** · October 5th 2009, 03:25 PM

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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