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Math Help - Find Unit vector parallel to xy plane and perp. to a vector

  1. #1
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    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
    <br />
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.
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  2. #2
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    Quote Originally Posted by JimmyRP View Post
    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
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  3. #3
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    so you got

    <br /> <br />
1x-2y+2(0)=0<br />

    <br /> <br />
x-2y=0<br />

    so

    <br /> <br />
x=2y<br />
    so the vector
    <br /> <br />
\left\langle {2y,y,0} \right\rangle<br />
    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|}
    =
    <br /> <br />
\left\langle {\frac{2y}{{\sqrt 5 }|y|},\frac{y}{{\sqrt 5 }|y|},0} \right\rangle<br />

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